Annual KSF Meeting November Protaras, Cyprus

Transcription

1 Annual KSF Meeting 2012 October th th 30 - November Protaras, Cyprus Proposed problems for KSF contest 2013

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3 Contents 1 PreEcolier point problems point problems point problems Ecolier point problems point problems point problems Benjamin point problems point problems point problems Cadet point problems point problems point problems Junior point problems point problems point problems Student point problems point problems point problems

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5 PreEcolier, 3 point problems 1 PreEcolier point problems What is the sum of the numbers covered by kangaroos? Pre-Ecolier (A) 20 (B) 50 (C) 100 (D) 110 (E) There are 12 books on a shelf and four children in a room. How many books will be left on the shelf if each child takes one book? (A) 12 (B) 8 (C) 4 (D) 2 (E) The rabbit is sitting in the right bottom corner of the maze. How many carrots can eat a rabbit walking in this maze? 5

6 PreEcolier, 3 point problems (A) 7 (B) 8 (C) 9 (D) 15 (E) 16 Pre-Ecolier Kasia has three sisters and three brothers. How many brothers and how many sisters her brother Mike has? (A) 3 brothers and 3 sisters (B) 3 brothers and 4 sisters (C) 2 brothers and 3 sisters (D) 3 brothers and 2 sisters (E) 2 brothers and 4 sisters At 7 o clock in the morning Tom has breakfast. His dinner is 11 hours later. What time is Tom s dinner? (A) 4 PM (B) 6 PM (C) 8 PM (D) 10 PM (E) 11 PM It is 10 o clock in the morning now and an hour has passed since Ann started her threehour train ride. What time will the ride end? (A) 3 PM (B) 2 PM (C) 1 PM (D) 12 PM (E) 11 AM Which of the dresses has less than 7 but not less than 6 dots? (A) (B) (C) (D) (E) Math Kangaroo 2013 is on March 21, 2013 and it is always in March. Adam is 7 years old. His birthday is in February. How old will Adam be in Math Kangaroo 2020? (A) 20 years old (B) 15 years old (C) 14 years old (D) 13 years old (E) 23 years old I am a number. If you add me to 30, you will get a number less than 60. If you count by tens you will say my name. I am not ten. Who am I? (A) 20 (B) 30 (C) 40 (D) 50 (E) 60 6

7 Pre-Ecolier PreEcolier, 3 point problems What number goes on the stone closest to the house? (A) 13 (B) 15 (C) 16 (D) 19 (E) This table shows the favorite toys of a group of children. Which toy was the favorite of the most children? (A) (B) Ann has Where is Barb? (A) (C) (D). Barb gave Eve (B). Jim has (C) (D) 7 (E). Bob has (E).

8 PreEcolier, 3 point problems In London 2012 USA won the most medals: 46 gold, 29 silver and 29 bronze. How many medals won USA alltogether? (A) 94 (B) 98 (C) 104 (D) 108 (E) 114 Pre-Ecolier A path is paved with square tiles. How many tiles fit in the area inside? (A) 5 (B) 6 (C) 7 (D) 8 (E) Which is the next? (A) (B) (C) (D) (E) Ann has twelve tiles: She makes one wire with these tiles. Ann starts at the left side. Which tile comes at the end of the arrow? (A) (B) (C) (D) (E) this is impossible The digits of the number 9145 are arranged in descending order and then in ascending order. The difference between the resulting numbers is: (A) 3726 (B) 8192 (C) 8182 (D) 8082 (E) Masha has 2 brothers and 2 sisters in her family. How many children are there in this family? (A) 2 (B) 3 (C) 4 (D) 5 (E) In the following figures white, grey and black kangaroes are represented. In what figure 8

9 PreEcolier, 3 point problems the number of black kangaroos is greater than the number of white ones? (A) (B) (C) Pre-Ecolier (D) (E) Anna likes to play with her doll and its clothes. If she has four different sweaters and two different skirts, in how many different ways can Anna dress her doll? (A) 2 (B) 4 (C) 6 (D) 8 (E) One litre of water weighs 1000 grams. What object contains 100 grams of water? (A) a glass (B) a jug (C) a bucket (D) a barrel (E) a tank The rectangular mirror was broken into some pieces. Which of the following pieces is missing in the given figure of this mirror? (A) (B) (C) (D) (E) Gaby baked a wonderful cake as a birthday present for Lotta. It looks like a woman. Lotta cuts a big piece. Which one? 9

10 PreEcolier, 3 point problems Pre-Ecolier (A) (B) (C) (D) (E) Theo s aunt baked cookies which are formed like digits (see picture). Which digits are missing? (A) 3 and 5 (B) 4 and 8 (C) 2 and 0 (D) 6 and 9 (E) 7 and When leaving for school today Flora was given by her mother 3 small apples each for Paul, Helen, Jim and herself. How many apples are this altogether? (A) 4 (B) 6 (C) 8 (D) 9 (E) The radius of a circle is increased by 120%, then the area of the circle is increased by: (A) 120% (B) 240% (C) 360% (D) 384% (E) 394% A zoo had a baby-boom: three lion cubs, two dolphins and four eaglets were born. How many more legs does the zoo count now? (A) 24 (B) 22 (C) 20 (D) 18 (E) Father has bought 5 apples for each of his four children. Karlo has given two apples to Ana, Ana has given three apples to Sanja and Sanja has given a half of her apples to Mihael. 10

11 PreEcolier, 4 point problems How many apples has Mihael got? (A) 5 (B) 6 (C) 7 (D) 8 (E) point problems A two inch long caterpillar has reached the shortest side of a brick that is 2 inches tall, 3 inches wide, and 6 inches long. It lies on the ground, perpendicular to the brick, as shown in the picture. The longest side of the brick faces the ground. Without changing its direction, at least how many inches does the caterpillar have to crawl in order to completely cross to the Pre-Ecolier opposite side of the brick? (A) 8 (B) 9 (C) 10 (D) 12 (E) Different answer Donkey collected blue flowers with red thorns. He noticed that each flower had 7 blue petals around its center and between any two neighboring petals there was a blue thorn, as shown in the picture. How many thorns did the flower have? (A) 3 (B) 2 (C) 6 (D) 7 (E) How many more bricks are on the larger stack? (A) 4 (B) 5 (C) 6 (D) 7 (E) At the park students need to plant 20 trees. They can plant 5 trees in 10 minutes and started at 9 in the morning. At what time will they finish with planting all 20 trees? (A) At 9:10 (B) At 9:20 (C) At 9:40 (D) At 9:50 (E) At 10: George has 2 cats of the same weight. What is the weight of one cat if George weighs 11

12 PreEcolier, 4 point problems Pre-Ecolier 25 pounds? (A) 1 pound (B) 1.5 pounds (C) 2 pounds (D) 2.5 pounds (E) 3 pounds In the game it is possible to make the following exchanges: Adam has 6 pears. How many strawberries will Adam have, when he trades all his pears only for strawberries? (A) 12 (B) 36 (C) 18 (D) 24 (E) Joseph is one year old. His father is 30 years older than Joseph. His mother is three years younger than his father. How old is Joseph s mother? (A) 31 (B) 30 (C) 29 (D) 28 (E) There is a blue house on each corner of the streets. There are 2 yellow houses on each street between the corner houses. How many houses are there in all? 12

13 PreEcolier, 4 point problems (A) 8 (B) 12 (C) 16 (D) 20 (E) Other answer Pre-Ecolier Brandon organizes his smiley face stickers in the following way: white, yellow, green, orange, red, purple, and again: white, yellow, green, orange, red, purple, etc. Which sticker will be the 21st in a row? (A) (B) (C) (D) (E) Ann has a lot of these pieces: She tries to put them in the square, as many as possible. How many cells shall be left empty? (A) 0 (B) 1 (C) 2 (D) 3 (E) Ann has a lot of these pieces: She tries to put them in the square, as many as possible. How many cells shall be left empty? (A) 0 (B) 1 (C) 2 (D) 3 (E) Ann has two parents and four grandparents. How many great-grandparents she has? (A) 4 (B) 6 (C) 8 (D) 10 (E) Cat and Mouse are moving to the right. When Mouse jumps 1 tile, Cat jumps two tiles. tile Cat catches Mouse? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 At which 13

14 PreEcolier, 4 point problems Pre-Ecolier There are more white than grey cells. How much more? (A) 8 (B) 9 (C) 16 (D) 17 (E) Which are the most? (A) (B) (C) (D) (E) all equal Which area is the biggest? (A) (B) (C) (D) (E) A cake is divided into 12 parts. Albert and Ben eat each 2 parts, Chris and Dee eat each 3 parts. How many parts remain? (A) 0 (B) 1 (C) 2 (D) 3 (E) How does the row of circles continue? (A) (B) (C) (D) (E) Ann sees a building from the front side like in the picture. What 14

15 PreEcolier, 4 point problems does she see from the back side? (A) (B) (C) (D) (E) something else When Pinocchio lies, his nose gets 6 cm longer. When he says the truth, the nose gets 2 cm shorter. When his nose was 9 cm long, he said one lie and one true sentence. How long was Pinocchio s nose afterwards? (A) 4 cm (B) 5 cm (C) 9 cm (D) 13 cm (E) 17 cm Pre-Ecolier Peter got so enthusiastic about watching the Olympics that he built the winners podium (as in the picture) at home. How many cubes did he need for it? (A) 12 (B) 16 (C) 18 (D) 22 (E) How many legs do one rooster, two sheep, three piglets, four ducks and five snakes have together? (A) 15 (B) 20 (C) 30 (D) 40 (E) There are 5 children in Kitty family. Kitty is 2 years older then her sister Betty, but 2 years younger then her brother Dannie. Teddy is 3 years older then Annie. Betty and Annie are twins. Who of them is most older? (A) Annie (B) Betty (C) Dannie (D) Kitty (E) Teddy John is organizing his photos in an album. He holds each photo using four stickers. If he wants to put 6 photos in each of the 50 pages of his album, how many boxes, with 50 stickers each, he will need to buy? (A) 4 (B) 6 (C) 24 (D) 30 (E) Ana has in her pocket one coin of 5 cents, one coin of 10 cents, one coin of 20 cents and one coin of 50 cents. How many different quantities can she form with these coins? (A) 4 cents (B) 7 cents (C) 10 cents (D) 15 cents (E) 20 cents A cat eats two mice per day. How many mice do it eat in two weeks? (A) 2 (B) 7 (C) 14 (D) 21 (E) Which of these five numbers is the largest? (A) 6 plus the double of 2 (B) 5 plus the double of 3 (C) 4 plus the double of 4 15

16 PreEcolier, 5 point problems (D) 3 plus the double of 5 (E) 2 plus the double of 6 Pre-Ecolier Among the distinct letters of the word, how many can be written so that the pencil does not leave the paper and no line is passed twice? (A) 1 (B) 2 (C) 3 (D) 4 (E) none Maja makes paper butterflies. She has made 4 of them on Monday. On Tuesday she has made three times more and on Wednesday she has made four times more than on Monday. How many butterflies has she made all together? (A) 19 (B) 22 (C) 30 (D) 32 (E) A pansy has got 5 petals and a snowdrop has 3 petals. How many petals have 6 pansies and 7 snowdrops got together? (A) 35 (B) 41 (C) 46 (D) 51 (E) There are 19 girls and twelve boys playing on the school playground. How many children would have to join them so they all could be divided into 6 equal groups? (A) 1 (B) 2 (C) 3 (D) 4 (E) point problems Sarah has cut some figures out of white paper. Three of the pieces of paper can be used to make a rectangle (without overlapping them). Which three? (A) R, G, T (B) T, M, P (C) T, G, N (D) R, G, N (E) M, R, P Nineteen students are going to sing in a choir. Each student receives a T-shirt from Mr. Pitt and a hat from Ms. Jolie, and then runs up on stage. Ms. Jolie hands out hats in this order: Red, Green, Blue, Red, Green, Blue and so on. Mr. Pitt hands out T-shirts in this order: Red, Red, Red, Green, Green, Green, Blue, Blue, Blue, Red, Red, Red, Green, Green, Green, Blue, Blue, Blue, Red, and so on. Thus, the first student goes up with a red hat and a red T-shirt, the second student goes up with a green hat and a red T-shirt. When all the students are up on stage how many of them have the same color T-shirt and hat? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 16

17 PreEcolier, 5 point problems How many squares are in the picture? (A) 15 (B) 14 (C) 11 (D) 10 (E) Three children draw three pictures in 3 minutes. How many pictures will six children draw in 6 minuts? (A) 6 (B) 12 (C) 14 (D) 16 (E) 18 Pre-Ecolier Four sunflowers and two roses cost 26 USD altogether. One sunflower costs 5 USD. How much does one rose cost? (A) 1 USD (B) 2 USD (C) 3 USD (D) 4 USD (E) 5 USD The tourist went for the three-day trip. At the second day he passed three kilometers more than at the first day. At the third day he passed two times kilometers more than at the second day. Total visitor passed 49 km. How many kilometers the tourist went over the first two days? (A) 26 (B) 39 (C) 23 (D) 10 (E) A green large cube is created from 27 little equal-sized cubes. Ania has removed four small cubes and she received the figure shown at the picture. Next she made stamps using the walls of this figure. How many of the following stamps can Ania receive? (A) 1 (B) 2 (C) 3 (D) 4 (E) In a park there are babies in four-wheel strollers and children on two-wheel bikes. Paula counted wheels and the total was 12. When she added the number of strollers to the number of bikes, the total was 4. How many two-wheel bikes are there in the park? (A) 1 (B) 2 (C) 3 (D) 4 (E) Other number A tile fall off the wall. Caroline has three extra tiles, see the picture. Which tiles fit? 17

18 PreEcolier, 5 point problems (A) a and b (B) a and c (C) b and c (D) only c (E) all three of them fit Pre-Ecolier Four rectangles built a square. Three of them are: What is the fourth rectangle? (A) (B) (C) (D) (E) With matchsticks Sophie made a row of 10 houses. In the picture you can see the beginning. How many matchsticks did Sophie need? (A) 40 (B) 41 (C) 45 (D) 52 (E) Three children were given 15 sweets. Ann eats 3 sweets. Boris and Caroline ate the rest, each the same. How many sweets did Boris ate? (A) 3 (B) 4 (C) 5 (D) 6 (E) Hugo eats 4 pieces of fruit everyday. His mum and dad eat 5 pieces each of them. How many pieces do they eat together during one week? (A) 14 (B) 45 (C) 63 (D) 70 (E) Squirrel Julie had 2 rooms with one nut in each, 2 rooms with 2 nuts in each, 2 rooms with 3 nuts in each, 2 rooms with 4 in each and 2 rooms with 5 nuts in each. She placed all the nuts to a few rooms in a way that in each there were 10 nuts. How many rooms did she fill like that? (A) 1 (B) 2 (C) 3 (D) 4 (E) According to pictures, which animal is the heaviest? 18

19 PreEcolier, 5 point problems Pre-Ecolier (A) (B) (C) 19

20 PreEcolier, 5 point problems Pre-Ecolier (D) (E) boy participated in a running competition: Andras, Bela, Csaba and Dezso. Andras was in the first 2, but Dezso not. Neither Csaba nor Dezso was the last one. Csaba was second or third. What was the result? (A) Bela, Csaba, Dezso, Andras (B) Andras, Bela, Csaba, Dezso (C) Bela, Andras, Csaba, Dezso (E) Andras, Csaba, Dezso, Bela (D) Andras, Dezso, Bela, Csaba A square box is filled with two layers of identical square pieces of chocolate. Kirill has eaten all 20 pieces located in the upper layer along the side walls of the box. How many pieces of chocolate are left in the box? (A) 16 (B) 30 (C) 34 (D) 52 (E) Peter has one brother, and his sister Kate has as many brothers, as sisters. How many children are there in that family? (A) 2 (B) 3 (C) 4 (D) 5 (E) How many triangles are there in the figure? (A) 8 (B) 9 (C) 12 (D) 15 (E) 16 20

21 Ecolier, 3 point problems 2 Ecolier point problems By drawing two circles, Mike obtained a figure, which consists of three regions (see pic.). At most how many regions could he obtain by drawing two squares? (A) 3 (B) 5 (C) 6 (D) 8 (E) A round pie can be divided into two equal parts by one straight cut and into three equal parts by three straight cuts. At least how many cuts are necessary to divide a round pie into twelve equal parts? (A) 4 (B) 6 (C) 8 (D) 10 (E) 12 Ecolier Alexandra has 3 pairs of trousers, 3 T-shirts and 3 scarves. What is the maximum number of outfits which she can get? (A) 9 (B) 6 (C) 27 (D) 100 (E) children from Adventure Park took part in competions. If 15 of them took part in the moving bridge contest, and 20 of them went down by tiroliana, how many children from Adventure Park took part in both events? (A) 5 (B) 15 (C) 30 (D) 10 (E) Monica arrived in the Kangaroo Camp on the 25th July in the morning and left the camp on 3rd August in the afternoon. How many nights did she sleep in the camp? (A) 7 (B) 9 (C) 10 (D) 30 (E) Find the rule behind the sequence. What is the next number? 1; 22; 111; 3333; 11111;... (A) (B) (C) (D) 10 (E) How many triangles can be seen in the picture below? (A) 12 (B) 13 (C) 14 (D) 15 (E) One day after my last birthday I could say The day after tomorrow is a Thursday. What weekday was my last birthday? (A) Monday (B) Tuesday (C) Wednesday (D) Thursday (E) Friday 21

22 Ecolier, 3 point problems Jill starts at K and jumps from one letter to the next. John starts at the 3 and jumps simultaneously from one character to the next in the opposite direction as shown in the figure. At what character are they simultaneously? (A) G (B) A (C) 2 (D) R (E) There is no such character Ecolier Franz is coloring geometric shapes. He has drawn a square. He has put little circles in each corner and thick lines on each side. He wants all circles and all lines to have different colors. How many colors does he need? (A) 2 (B) 6 (C) 8 (D) 12 (E) Counting the train cars from the locomotive to the last car, Sara realizes she has a reserved seat in the 21st car. Counting the train cars from the last car to the locomotive, she realizes she has the same reserved seat in the 7th car. How many cars does the train have? (A) 28 (B) 25 (C) 27 (D) 29 (E) In London 2012 USA won the most medals: 46 gold, 29 silver and 29 bronze. On the second place was China with 38 gold, 27 silver and 23 bronze. With how many medals won USA more? (A) 6 (B) 14 (C) 16 (D) 24 (E) Ann walks in the direction of the arrow. First she goes to the right, then to the left then again to the left, then to the right then to the left, and finally again to the left. Then Ann has walked towards (A) a (B) b (C) c (D) d (E) e On a road these equilateral triangles indicate that you have to give way to the traffic on a crossing road. The space between two of these triangles is as long as the size of a triangle. We look at seven of these triangles and the part of the road between them (grey in the picture). How many percent in this picture is triangle? (A) 20 (B) 25 (C) 28 (D) 32 (E) 37 22

23 Ecolier, 3 point problems ? + 5? = 104. Which digit must be substituted for the question mark to make the calculation correct? (A) 2 (B) 4 (C) 5 (D) 7 (E) Maria and Jane were collecting strawberries. Maria collected 21, Jane three times as many as Maria. As they are good friends, they divided all the strawberries equally in two halves. How many strawberries had each one of them? (A) 12 (B) 24 (C) 32 (D) 42 (E) Daniel had a package of 36 candies. He divided all the candies equally among all his friends. How many friends he could not have there for sure? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 Ecolier Justin built up an object from cubes. How many cubes did he use? (A) 7 (B) 8 (C) 9 (D) 10 (E) When Pinocchio lies, his nose gets 6 cm longer. When he says the truth, the nose gets 2 cm shorter. When his nose was 9 cm long, he said three lies and two true sentences. How long was Pinocchio s nose afterwards? (A) 14 cm (B) 15 cm (C) 19 cm (D) 23 cm (E) 31 cm Eight regular dice are clued together to form a cube. What is the maximum of pips it can be on one side? (A) 24 (B) 23 (C) 21 (D) 20 (E) In London 2012 in the group stage the american men basketball team gets 589 point in 5 games. If the ball fall in to basket, the team get 1, 2 or 3 point. Minimum how many time was thrown the ball by US-players in to basket in this 5 games? (A) 195 (B) 196 (C) 197 (D) 198 (E) Misha measured the lengths of 5 sticks and wrote down the results. Which of his results is the largest? (A) 3 cm (B) 3 dm 2 cm (C) 35 mm (D) 3 cm 7 mm (E) 302 mm About the number 325, five boys said: Andrei: This is a 3-digit number ; Boris: All digits are distinct ; Vitya: The sum of the digits is 10 ; Grisha: The digit of units is 5 ; Danya: All digits are odd. Who of the boys was wrong? (A) Andrei (B) Boris (C) Vitya (D) Grisha (E) Danya 23

24 Ecolier, 3 point problems Peter travelled from Lisbon to New York. The flight departed from Lisbon at 12:30 (local time) and arrived in New York at 15:40 (local time). If in New York is 5 hours earlier than in Lisbon, which was the duration of the flight? (A) 1h50m (B) 3h10m (C) 5h (D) 7h10m (E) 8h10m The letters of the word KANGAROO have been written to the grid like has been showed in the diagram. Let say that two squares are adjoining squares if they have the same point. Which answer below represents the grid where no letter has been placed neither in the same Ecolier nor adjoin square compared with the initial diagram? (A) (B) (C) (D) (E) In the following figures white, grey and black kangaroes are represented. In what figure the number of black kangaroos is greater than the number of white ones? (A) (B) (C) (D) (E) The rectangular mirror was broken into some pieces. Which of the following pieces is missing in the given figure of this mirror? (A) (B) (C) (D) (E) 24

25 Ecolier, 3 point problems Paula decided to order her friends on the playground by height alphabetically. What is the correct order? (A) Alice, David, Emma, Hannah, John (B) Alice, Emma, Hannah, David, John (C) John, Hannah, Emma. David, Alice (E) Unable to determine (D) David, John, Alice, Emma, Hannah When climbing, you have to follow the three-point rule, which means the climber is supposed to maintain three limbs in contact with the rock. Which option below is incorrect? (A) foot, hand, hand (B) hand, foot, hand (C) foot, foot, hand (D) hand, hand, hand (E) hand, foot, foot Jonas, Vera, Alina and Tom are on a bicycle tour. After they climbed a steep mountain Alina opens a bag with sweets. She counts 16 sweets and gives the same number of sweets to each of the bikers including herself. How many sweets does each biker get? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 Ecolier There are three families in my neighborhood with 3 children each, two of these have twins. All twins are boys. How many girls are in these families at most? (A) 2 (B) 3 (C) 4 (D) 5 (E) When coming home my siblings and me arrange our shoes in front of the door. Eva s shoes are smaller than Jan s but bigger than David s. Lisa s are bigger than mine but smaller than David s. Who has the biggest shoes? (A) Eva (B) Jan (C) David (D) me (E) Lisa Rmi arrived 25 minutes late. Jules arrived just in time at 15:45. Quentin arrived a quarter of an hour before Rmi. At what time did Quentin arrive? (A) 15:05 (B) 15:45 (C) 15:55 (D) 16:05 (E) 16: Which of the following has fewer axes of symmetry? (A) A (B) B (C) C (D) D (E) E Sue was born on the 5th of March, 1998, and Robert was born on the 15th of December, Which of the following is true? (A) Sue is 11 years older than Robert. (B) On the 5th of January, 2004, Sue was 6 years old. (C) On the 5th of January, 2004, Robert was 16 years old. (D) When Sue was born, Robert was 11 years old. (E) None of the above. 25

26 Ecolier, 3 point problems How many odd numbers are there between 9 and 22? (A) 8 (B) 7 (C) 6 (D) 5 (E) boys came to John s 10th birthday party. Each boy takes a seat around the round table so that between each two boys sits one of the girls who have just arrived. At least how many pieces to cut the cake so that each receives at least one piece? (A) 7 (B) 10 (C) 11 (D) 14 (E) Mother has bought 21 apples. Ivana has eaten one third of the apples, Ana has eaten eight and Ante the rest of the apples. How many apples has Ante eaten? (A) 3 (B) 9 (C) 11 (D) 6 (E) 7 Ecolier Mother has baked a cake for Petra and she has cut it in 24 equal pieces. Father has eaten one half of the cake, brother has eaten two pieces and Petra three pieces. How many pieces of the cake is left? (A) 3 (B) 10 (C) 7 (D) 9 (E) If Frank was in the bakery bought two rolls, left him two coins, but if he bought three rolls he owes three coins. What is the price of the one roll? (A) 2 (B) 3 (C) 4 (D) 5 (E) I write the sum of three natural numbers greater than 0, and the result is Which is the major possible of one of these numbers? (A) 571 (B) 671 (C) 2012 (D) 2010 (E) How many natural numbers between 1 and 99 can be written using only one digit 3? (A) 99 (B) 70 (C) 19 (D) 18 (E) In his shop, Pedro has three kind of boxes to pack 5 oranges, to pack 9 oranges or to pack 10 oranges. If he wants to pack 48 oranges, which is the least quantity of boxes he can use? (A) 8 (B) 7 (C) 6 (D) 5 (E) At Anna set up an alarm clock to wake her up after 10 hours and 20 minutes. Next morning she woke up 40 minutes before the alarm. When did Anna wake up? (A) 5.55 (B) 6.15 (C) 6.35 (D) 6.55 (E) Carolin s mother has 2 brothers and 3 sisters, each of them has at least one child. Carolin has 3 male cousins and 4 female cousins. At most how many sisters can have Carolin s female cousin? (A) 1 (B) 2 (C) 3 (D) 4 (E) In the block of flats with three floors there are four flats in each floor. Each flat has 6 windows. There are additional 2 windows in the corridor of each floor, except in the first floor where there is only one window. How many windows does the building have? (A) 35 (B) 77 (C) 83 (D) 95 (E)

27 Ecolier, 3 point problems Hairdresser Luke needs 30 minutes to make a hairstyle for a man, and 1 hour and a half to make a hairstyle for a woman. One day, three women made an appointment at the Hiardresser. How many men can Luka appoint in addition, if he will work for 6 hours? (A) 1 (B) 3 (C) 6 (D) 11 (E) There are 13 trees in a row. If we keep the first tree and cut the second down, the fourth, the sixth, etc. How many trees remain standing? (A) 8 (B) 7 (C) 6 (D) 9 (E) 4 Ecolier There are 13 trees in a row. If we keep the first tree and cut the second down, the fourth, the sixth, etc. How many trees were cut down? (A) 8 (B) 7 (C) 6 (D) 9 (E) Techi s storybook has 20 leaves. His little brother took out 3 leaves of her book. How many pages remain in the book? (A) 34 (B) 32 (C) 30 (D) 28 (E) Vero s mum prepares cheese and ham sandwiches. A package of bread has 24 slices. How many sandwiches can she prepare with two and a half packages? (A) 24 (B) 30 (C) 48 (D) 34 (E) Which of the following pieces covers the largest number of dots in the table? (Piece must lie completely on the table.) 27

28 Ecolier, 4 point problems (A) (B) (C) (D) (E) point problems Ecolier How many four-digit numbers have the sum of the digits 35? (A) 3 (B) 0 (C) 4 (D) 1 (E) Follow the algorithm for addition the elements of the series. Find the number that the symbol? replaces. (A) 0 (B) 6036 (C) 8048 (D) 6012 (E) If {[(a + 1) 2] 3} : 4 = 6, then a + a a + a a a + a a a a =... (A) 7380 (B) 7480 (C) 8480 (D) 738 (E) Find the sum of all the numbers which divided by 10 give 10 as the quotient. (A) 2045 (B) 1054 (C) 100 (D) 1045 (E) One of my grandmothers has two daughters and the second grandmother has two daughters and one son. Each of my aunts has two children. Me, my two siblings and all children of my aunts are playing together outdoors. How many children are playing outdoors? (A) 8 (B) 9 (C) 10 (D) 11 (E) In a book each chapter begins on the new page and after the end of each chapter there is one page with illustration. Chapter 2 consist of 22 pages, chapter 3 of 26 pages and chapter 4 of 21 pages. On which page does chapter 5 begin if we know that chapter 2 begins on page number 24? (A) On page number 92. (B) On page number 94. (C) On page number 93. (D) On page number 96. (E) On page number John has thought of a three-digit number in which the sum of all digits is equal to 10. He has multiplied it by 9 and become a three-digit number. What number has John thought of? (A) 208 (B) 307 (C) 109 (D) 406 (E) Hanna is telling Abed about numbers. She is thinking about a certain number. She says that this number together with half of the number is 24. What number is she thinking about? (A) 2 (B) 4 (C) 12 (D) 16 (E) 18 28

29 Ecolier, 4 point problems Dad has a basket of marbles. The number of marbles is less than 20. If Dad divides the marbles amongst four children, 3 marbles are left. If Dad divides the marbles amongst three children, 2 marbles are left. If Dad divides the marbles amongst two children, 1 marble is left. How many marbles are there in the basket? (A) 7 (B) 11 (C) 15 (D) 17 (E) Using 15 matchsticks Sophie made a row of 7 triangles. How many matchsticks will Sophie need to make such a row of 21 triangles? (A) 40 (B) 41 (C) 42 (D) 43 (E) After the Olympics last year in a certain country the athletes were paid some money for each medal. Albert won 2 silver and 1 bronze medal and received EUR. Ben won 1 silver and 2 bronze medals, he received EUR. Carl won 1 bronze and 2 gold medals and got EUR paid. Damian won 2 bronze and 1 gold medal. How much money did he receive? (A) EUR (B) EUR (C) EUR (D) EUR (E) EUR Ecolier Ann has turned around a building block like the one in the picture at the right hand. She puts it exactly before the one in the picture. How does the result look like? (A) (B) (C) (D) (E) Father is 33 years old. His three sons are 5, 6 and 10 years old. In how many years will the three sons together be as old as their father? (A) 4 (B) 6 (C) 8 (D) 10 (E) Children in the school club had to arrange fitness balls according to their size from the biggest to the smallest one. Rebecca was comparing them and said: the red ball is smaller than the blue one, the yellow one is bigger than the green one, the green one is bigger than the blue one. What is the correct order of the fitness balls? (A) green, yellow, blue, red (B) red, blue, yellow, green (C) yellow, green, red, blue 29

30 Ecolier, 4 point problems (D) yellow, green, blue, red (E) blue, yellow, green, red Pirates shipped out of the island to the Mediterranean sea at exact noon. They sailed for 96 hours. How many nights did they spend at sea? (A) Once. (B) Twice. (C) Three times. (D) Four times. (E) Five times The winner of the Big Snail Race was the snail that covered the longest distance in three hours. The judges measured these distances: snail A: 2 m 6 dm 3 cm 2 mm, snail B: 260 cm 23 mm, snail C: 26 dm 2 cm 3 mm, snail D:2 m 3 dm 62 cm, snail E: 2 m 6 dm 63 mm. Which snail was the winner? (A) A (B) B (C) C (D) D (E) E Ecolier The princess got a necklace with 25 jewels - sapphires, rubies, emeralds and amethysts. There were 8 sapphires. There were 3 less rubies than sapphires. The number of emeralds and amethysts in the necklace was the same. How many amethysts were there? (A) 5 (B) 6 (C) 8 (D) 12 (E) Leo writes in arrow the number 1, 2, 3,..., 2012, Which number is in the middle position? (A) 1005 (B) 1006 (C) 1007 (D) 1008 (E) The number 36 has the property that it is divisible by the digit in the unit position, because 36 divided by 6 is 6. The number 38 does not have this property. How many numbers between 20 and 30 has this property? (A) 2 (B) 3 (C) 4 (D) 5 (E) Joining the mid points of the sides of a first square we obtain a second square. We repeat this until the fifth square. How many of the small squares fit in the first one? (A) 5 (B) 8 (C) 10 (D) 16 (E) There is a track of competitions which is made out of rectangles m. A kangaroo should move from the start to the finish so that she passes points A, B and C in this order and being no more than one time at each of these three points. How many meters should the kangaroo pass at least? (A) 170 m (B) 180 m (C) 190 m (D) 200 m (E) 210 m Kangaroos K, L, M, N and O stand on line and make jumps in this order. Every kangaroo reaches always with one jump to next free square. In which order are the kangaroos from left to right when K and L have made 3 jumps, M and N 2 jumps and O one jump? (A) OKMLN (B) OKLMN (C) OMLKN (D) OMLNK (E) OMKLN 30

31 Ecolier, 4 point problems There are two discs with the same centre and the shaded disc is rotatable. It can be rotated so that after rotating the lines from centre of both discs overlap. On the picture there is no sector where letters coincide on shaded and unshaded disc. Find the biggest number of sectors where letters on shaded and unshaded discs coincide after rotating. (A) 2 (B) 3 (C) 4 (D) 5 (E) Mary was born on March 15th Today, on March 15th 2013 she wrote How many digits were in the row? (A) 180 (B) 189 (C) 191 (D) 198 (E) Andy, Betty, Cathie and Dannie were born on 20/02/2003, 12/04/2003, 12/05/2003 and 25/05/2003 (day/month/year). Betty and Andy were born in the same month. Andy and Cathie were born in the same day of different months. Who of these children is the oldest? (A) Andy (B) Betty (C) Cathie (D) Dannie (E) impossible to determine Ecolier Jamie s father tiled the rectangular bathroom floor with square tiles as shown in the picture. For the boundary he had to cut some tiles in half, but fortunately only a few. How many? (A) 3 (B) 4 (C) 6 (D) 7 (E) Ella wants to build a cube using bricks that are 3 cm long, 2 cm high and 1 cm wide. How many are needed if the cube should be as small as possible? (A) 6 (B) 9 (C) 12 (D) 20 (E) Which of the following pieces fits to the piece on the right picture such that together they form a rectangle? (A) (B) (C) (D) (E) Nick wrote out the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. He underlined 3 odd numbers, and then underlined 2 other numbers divisible by 3. After that, he found the sum of the 5 31

32 Ecolier, 5 point problems numbers underlined. What is the smallest number he could obtain? (A) 15 (B) 20 (C) 22 (D) 24 (E) In December Tosha-the-cat has slept exactly 3 weeks. How many minutes did he stay awake? (A) (31 7) (B) (31 7 3) (C) (30 7 3) (D) (31 7) (E) (31 7 3) After 2013, how many years will pass before the following event happens for the first time: the product of digits in the notation of the year is greater than the sum of these digits? (A) 87 (B) 98 (C) 101 (D) 102 (E) 103 Ecolier Mother bought 17 cones of ice-cream for her three children. The number of cones eaten by Misha is twice the number of those eaten by Masha. Dasha has eaten more ice-cream than Masha but less than Misha. How many cones of ice-cream have been eaten by Dasha? (A) 4 (B) 5 (C) 6 (D) 7 (E) John, Mark and George live in houses of different colours: red, blue and white. One of them plays football, the other plays basketball and the third plays tennis. We know that John lives in the red house, Mark is the one who plays tennis and that the basketball player lives in the white house. Which house does George live in and which sport does he play? (A) the red house, tennis (B) the blue house, tennis (C) the white house, tennis (D) the white house, basketball(e) the red house, football Three friends Ana, Iva and Tea have decided to buy new dresses. Each one of them has bought a dress in her favourite colour. Ana and Iva haven t bought the red dress. The one who was finished with the shopping first likes green. The one who was finished with the shopping last likes white but she isn t Ana. Iva isn t the second. Who was the first one to finish? Who was the second one to finish? Who was the third one to finish? From the first to the last. (A) Ana, Iva, Tea (B) Iva, Ana,Tea (C) Ana, Tea, Iva (D) Tea, Iva, Ana (E) Iva, Tea, Ana point problems All 4-digit positive integers with the same four digits as in the number 2013 are written in one line without gaps and in an increasing order. How many times does the pattern 01 appear in this sequence of digits? (A) 3 (B) 4 (C) 5 (D) 6 (E) ( ) : ( ) ( ) : [(15 15) : (3 3)+2 2 5] =... (A) 1 (B) 10 (C) 0 (D) 11 (E) Cristi has to sell 10 glass bells that vary in size: 1 euro, 2 euro, 3 euro, 4 euro, 5 euro, 6 euro, 7 euro, 8 euro, 9 euro, 10 euro. In how many ways can you divide all the glass bells in three packets so that all the packets have the same price? (A) 2 (B) 0 (C) 3 (D) 4 (E) 1 32

33 Ecolier, 5 point problems There are apples, apricots and peaches in a big basket. If there are 18 peaches and apricots, 30 fruits are not apricots and 28 are apples and apricots, how many fruits are there in the basket? (A) 46 (B) 20 (C) 40 (D) 38 (E) At a bank counter you can pay various sums of money using an identical number of banknotes: 5, 10, 20, 50 euro. How many such sums are expressed by three-digit numbers? (A) 8 (B) 9 (C) 11 (D) 10 (E) Bogdan has a puzzle composed of identical squares whose size length is 1 dm (see the picture). If the puzzle continues with three more rows of squares of the same size (following the pyramid-shaped structure), then the perimeter of the figure thus obtained is (A) 340 cm (B) 430 cm (C) 200 cm (D) 400 cm (E) 34 cm Ecolier In a chest of drawers there are four drawers (see the picture). In each of the drawers there are one of the following things: a hat, a scarf, gloves, an umbrella. Gloves are lying lower than scarf. Umbrella is lying lower than hat and than gloves. Scarf is not lying in the highest drawer. Which of the pictures below does present this chest of drawers? (A) (B) (C) (D) (E) A network of squares 8 5 can be built from pieces of a single type (see example above). Which of the following types of components can the network be built with? (A) (B) (C) (D) (E) 33

34 Ecolier, 5 point problems A rectangular path counts 100 tiles of 1 1 meter. How long is the outer edge? (thick in the picture) (A) 96 m (B) 98 m (C) 100 m (D) 102 m (E) 104 m Ecolier At an election with five candidates all of them got another number of votes. 36 votes were given, no abstentions. The winner got 12 votes, the number last got 4 votes. How many votes got the number second? (A) 8 (B) 8 or 9 (C) 9 (D) 9 or 10 (E) A train of length 400 m enters with a velocity of 400 m/min a tunnel with length 400 m. How many minutes later is the train out of the tunnel? (A) 1 (B) 2 (C) 4 (D) 5 (E) Ann has a lot of pieces like the one in the picture. How much of these pieces do you need at least to make a square? (A) 2 (B) 4 (C) 6 (D) 8 (E) that is impossible Iryna multiplies numbers by 5, Daryna adds 4, Volodya subtracts 3 and Petryk divides by 2. In what order do they need to perform their operations to get 1 from 11? (A) Iryna, Daryna, Volodya, Petryk (C) Iryna, Petryk, Volodya, Daryna (E) Volodya, Daryna, Petryk, Iryna (B) Iryna, Volodya, Daryna, Petryk (D) Daryna, Iryna, Volodya, Petryk A garden architect made a plan for tree planting in the rectangular park: plant one poplar in each park corner, plant seven pines to the centre of the park, along the border there shall be four lime-trees in-between every two poplars. How many trees are they going to plant according to this plan? (A) 16 (B) 23 (C) 27 (D) 34 (E) There were 60 cats in the yard. 35 of them had stripes, 33 had one black ear. At least how many cats had stripes and one black ear for sure? (A) 2 (B) 8 (C) 25 (D) 27 (E) 33 34

35 Ecolier, 5 point problems The palace has four floors, each floor has 5 rooms and in each room there are 6 windows. During the birthday celebration of the princess there were 2 flags in every second window. How many flags in total were there in the windows? (A) 15 (B) 60 (C) 80 (D) 120 (E) The digital clock in the room showed 20:13 when the chess game started. It was finished 20 hours and 13 minutes later. Then the digital clock showed (A) 20:13 (B) 16:13 (C) 15:26 (D) 16:26 (E) 20: In a camp 3 sportmen was talking eachother: Andras, Balazs and Csaba. They sportart are running, swimming and gymnastic. They are from 3 different city. The name of the swimmer from Debrecen is not Csaba. Csaba is not from Pecs. Balazs is not a runner and he is not from Gyor. The boy from Gyor is not a runner. What is the sportart of Andras and where is he from? (A) gymnastic, Pecs (B) running, Pecs (C) running, Gyor (D) swimming, Debrecen (E) gymnastic, Debrecen Ecolier In the number 2013 the sum of the first three digits is equal to the fourth digit. In the future years, how many times till 2999 the notation of the year will possess the same property? (A) 6 (B) 13 (C) 18 (D) 25 (E) Using the digits 1, 3, 5, 7 and 9 Eddy made three various two-digit numbers (any of the digits is used exactly once). To what of the following values the sum of these three numbers cannot be equal? (A) 89 (B) 201 (C) 185 (D) 213 (E) How many neighbour couples of months with an equal number of days are there in the period ? (A) 7 (B) 9 (C) 11 (D) 13 (E) Peter bought a carpet 36 dm long and 60 dm width. The carpet is composed, as can be seen in the figure, of small square containing either a sun or a moon. You can see that along the width there are 9 squares. When the carpet is fully deployed, how many moons there will be? 35

36 Ecolier, 5 point problems (A) 75 (B) 67 (C) 65 (D) 63 (E) Many guests came to Nadya s birthday party. More than half of them were boys, and the name of more than one third of the boys was Fedya. Altogether, there were three Fedya s. What is the largest possible number of guests that could have been at Nadya s birthday party? (A) 12 (B) 13 (C) 14 (D) 15 (E) Vasya wants to make several cards with digits (one digit on each card) so that with them he could compose any collection of four distinct numbers between 1 and 300. What is the smallest number of cards he needs to make? (A) 20 (B) 60 (C) 70 (D) 74 (E) 82 Ecolier Basil has several domino tiles drawn in the figure. He wants to arrange them in a line according to the following domino rule : in any two neighboring tiles, the neighboring squares must have the same number of points. What is the largest number of tiles he can arrange in this way? (A) 3 (B) 4 (C) 5 (D) 6 (E) Baby Roo can write only two digits: 0 and 1. He wrote down several numbers the sum of which is It turned out that it is impossible to get the same sum with a smaller number of summands of this sort. How many numbers were written by Baby Roo? (A) 2 (B) 3 (C) 4 (D) 5 (E) Mary has six shapes (see the figure). She wants to use all of them but one to compose a square and a rectangle. What shape will not be used if one of the rectangle s sides must be 6 cells long? (A) (B) (C) (D) (E) this situation is impossible Basil cut a wooden cube into identical smaller cubes and arranged the smaller cubes in a straight line of length 200. How long is the side of the initial cube if the side length of the smaller cube is an integer? (A) 5 (B) 10 (C) 25 (D) 50 (E) Simon counts the letters of his name the right way and back again, over and over, as indicated. Number 17 is in this way given to the letter S. Which letter is given to number 99? 36

37 Ecolier, 5 point problems (A) S (B) I (C) M (D) O (E) N Which of the following equalities, in each of which appear once each of ten digits, 0 through 9, is correct? (A) ( ) = 2013 (B) ( ) = 2013 (C) ( ) ( ) = 2013 (D) = 2013 (E) None of the above is correct Ecolier In a test in which the maximum was 10 points, the average of 10 students was 9,2 points. What is the lowest note that could get one of 10 students? (A) 2 (B) 9 (C) 9,2 (D) 4 (E) 0 37

38 Benjamin, 3 point problems 3 Benjamin point problems If a b = c and c d = 5 then a 2bd is: (A) 5 (B) 10 (C) 25 (D) 32 (E) other The sum of the prime factors of 2013 is (A) 75 (B) 94 (C) 194 (D) 674 (E) other Find the sum of all the proper fractions with denominator 13 (A) 5 (B) 7 (C) 6 (D) 3 (E) Find the distance which Mara covers to get to her Bunica. (A) 300 m (B) 400 m (C) 800 m (D) 1 km (E) 700 m Benjamin If half of a quarter third of a number is 24, then that number is equal to.... (A) 56 (B) 567 (C) 576 (D) 288 (E) The product of a two identical digit number aa and the digit a is 176. Find the value of digit a. (A) 6 (B) 4 (C) 11 (D) 44 (E) How many meters of fence do I need to surround the two lands? (A) 420 m (B) 300 m (C) 1000 m (D) 360 m (E) 400 m How many 4-digit numbers width different figures one can form using the digits of the number 2012? (A) 20 (B) 24 (C) 18 (D) 8 (E) The sum of the square of squares of the numbers 3 and 12, and the square of their sum is equal to (A) 600 (B) 38 (C) 120 (D) 194 (E) The family Gurukan has three little Gurukan-daughters. Every daughter has 2 Gurukanbrothers. How many people live in the Gurukan family? (A) 5 (B) 7 (C) 8 (D) 9 (E) 11 38

39 Benjamin, 3 point problems Anna lights every 10 minutes a candle. Every candle burns for 40 minutes. How many candles are alight 55 minutes after lighting the first candle? (A) 2 candles (B) 3 candles (C) 4 candles (D) 5 candles (E) 6 candles Andra, following a certain algorithm, has transformed a number until she got the number Find the algorithm and the number of steps needed for this transformation. (A) 670 (B) 671 (C) 669 (D) 500 (E) A container filled with water weighs 31 kg, and the same container half-filled with water weighs 17 kg. How much does the empty container weigh? (A) 1 (B) 2 (C) 3 (D) 4 (E) What is the minimum distance to be covered in order to get through the maize from A to B? (A) 50 squares (B) 30 squares (C) 55 squares (D) 54 squares (E) 53 squares Benjamin How many different squares can you identify in the picture? (A) 27 (B) 25 (C) 15 (D) 19 (E) Peter had the choice of sitting in two railway carriages recently. Both had 76 seats, but one had an empty mass of tonnes, the other tonnes. What is the difference in their masses? (A) tonnes (B) tonnes (C) tonnes (D) tonnes (E) tonnes Which of the following numbers is the largest? (A) (B) (C) (D) (E) The sum of the ages of Ann, Bob and Chris is 31 years. What will be the sum of their ages in three years time? 39

40 Benjamin, 3 point problems (A) 32 (B) 34 (C) 35 (D) 37 (E) What is the smallest positive integer which is the sum of three consecutive positive even numbers and also the sum of two consecutive odd numbers? (A) 8 (B) 12 (C) 18 (D) 24 (E) A watch of the expensive trade mark MeisterSinger has only one arm. What is the time shown in the picture? Benjamin (A) ten minutes before ten (B) five minutes before ten (C) five minutes after ten (D) ten minutes after ten (E) fifteen minutes after ten The picture shows a 3 3 slide puzzle. The piece in the left upper cell with the K must be moved to the right under cell. The right upper cell is empty. What is the minimal number of moves to achieve this? (A) 10 (B) 11 (C) 12 (D) 13 (E) it is not possible Ann has a lot of pieces like the one in the picture. She tries to punt as many as possible in the 6 by 6 square. What is the largest possible number of pieces Ann can put in the square? The pieces may not overlap each other. (A) 6 (B) 7 (C) 8 (D) 9 (E) 10 40

41 Benjamin, 3 point problems A square and a circle have equal perimeters. The ratio of the area of the square to the area of the circle is: (A) 1 : 1 (B) π : 1 (C) π : 2 (D) π : 4 (E) 1 : How many 3 2 rectangles are in the diagram? (A) 3 (B) 8 (C) 9 (D) 14 (E) There are two little fleas sitting on the number line: Tim and Tom. Tim sat on number 2, Tom on number 32. Another flea Molly came and sat down exactly in the middle between them. On which number did Molly sit? (A) 14 (B) 15 (C) 16 (D) 17 (E) 18 Benjamin Nathalie wanted to build the same cube as Diana had (picture 1). However, Nathalie ran out of small cubes and built only the part of the cube, as you can see in the picture 2. How many cubes is she missing? (A) 5 (B) 6 (C) 7 (D) 8 (E) Which of the shapes does not have the same perimeter as the grey square? (A) (B) (C) 41

42 Benjamin, 3 point problems (D) (E) Michael got special drops. First four doses had to be taken every 15 minutes. He took the first dose at 11:05. What time did he take the last dose? (A) 11:40 (B) 11:50 (C) 11:55 (D) 12:00 (E) 12: Max, David and Peter competed in ice cream eating. Max ate 6 scoops in the same time as David ate 4 scoops and Peter 5 scoops. Max ate 18 scoops in half an hour. How many scoops in total did they eat in half an hour together? (A) 27 (B) 30 (C) 33 (D) 45 (E) 51 Benjamin At the 2013 story tower, an elevator works in a very specific way; it takes passengers only 18 stories up and then 7 stories down, always in this pattern. After how many such cycles one can reach the 2013th floor of the Kanga tower? (A) 11 (B) 33 (C) 99 (D) 183 (E) it is impossible to get to the 2013th level in such way There are given four angles in one obtuse and one acute triangle: 130, 80, 55, and 10. What is the measure of the smallest angle of the acute triangle? (A) 15 (B) 25 (C) 30 (D) 40 (E) white and 135 brown eggs have to be packed separately in several egg containers. The containers have the same number of dimples and there were no empty dimples in any container and neither egg was put on the top of another. What is the largest possible number of dimples in one such container? (A) 5 (B) 9 (C) 15 (D) 27 (E) A die is a cube with its faces numbered from 1 to 6 (with dots). Numbers on opposite faces add up to 7. A die lays on a plane as shown in the figure. A move consists of a rotation of 90 degrees around one of the edges of the lower face, until a new face touches the plane. There are four possible moves named N, W, S and E, according as the die moves to the North, West, South or East, respectively. If we perform in sequence the moves N, W, S and E, how many points are there on the upper face in the final 42

43 Benjamin, 3 point problems position of the die? (A) 1 (B) 2 (C) 3 (D) 5 (E) other The number 36 has the property that it is divisible by the digit in the unit position, because 36 divided by 6 is 6. The number 38 does not have this property. How many numbers between 20 and 30 has this property? (A) 2 (B) 3 (C) 4 (D) 5 (E) Dimitar has a rectangular parallelepiped with lengths of the edges 5 cm, 5 cm and 9 cm. He wants to divide it into unit cubes and make smaller cubes from them, each of them built with more than one cube and all of them having different edges. What it the maximal number of cubes that Dimitar can build? (A) 0 (B) 1 (C) 2 (D) 3 (E) Ana thought on a natural number. She divided it by 2, then took the square root and afterwards multiplied it by 6, an obtained the same number she started with. In which number has Ana thought? (A) 3 (B) 9 (C) 12 (D) 18 (E) From the two numbers 2013 and 3014 we obtain Which number can we obtain in the same way from the numbers 3524 and 4528? (A) 2445 (B) 2454 (C) 3528 (D) 4245 (E) How many natural numbers of 3 digits exist with all digits even? (A) 64 (B) 80 (C) 100 (D) 125 (E) 130 Benjamin The difference between two numbers is 44. Increasing 5 units each, the largest is five times the lowest. What is the lowest number? (A) 4 (B) 5 (C) 6 (D) 7 (E) A student get 5 points if his exercise is correct and looses 3 points if his exercise is incorrect. After 50 exercises he has 130 points. The number of correct exercises is (A) 35 (B) 30 (C) 25 (D) 20 (E) Today, on March, 21th, 2013, Kangaroo contest takes place. The previous Kangaroo contest took place on March, 15th, How many days ago it was happened? (A) 365 (B) 359 (C) 350 (D) 370 (E) From a wooden cube with side 3 cm we cut at the corner a little cube with side 1 cm (see picture). What is the number of faces of the solid after cutting off such small cubes at each corner of the big cube? 43

44 Benjamin, 3 point problems (A) 16 (B) 20 (C) 24 (D) 30 (E) Mary had built a row of seven dominoes following instructions game (cells in contact have the same number of points). She knows that counting all the cells had been were 33 points. But his brother has released two pieces off the row. How many points were in the cell with the big question mark? (A) 1 (B) 2 (C) 3 (D) 4 (E) Paul s school is 454 years old in In what year was his school founded? (A) 1669 (B) 1641 (C) 1559 (D) 1561 (E) What is the least possible number of males in a family in which forty per cent of the family members are female? (A) 1 (B) 2 (C) 3 (D) 6 (E) What is the difference between the smallest and the largest of the following calculations? Benjamin (A) 4 (B) (C) 20 (D) 23 (E) The incorrect statement = 22 may be made correct by increasing one of the numbers in it by 1. Which number? (A) 1 (B) 3 (C) 6 (D) 2 (E) What is the value of 2 x + x 2 for x = 1? (A) 1 (B) 1.5 (C) 2 (D) 2.5 (E) Which of the 4 shapes on the right can be built by glueing together some sides of wooden pieces which have the shape of the piece shown on the left? (A) none (B) only D (C) B and D (D) A, B and D (E) all of them Ann, Bruno, Cindy and Daniel are sitting around a table; each has a hat with their name. They exchange hats by turns, as follows: First Ann exchanges her hat with Bruno, 44

45 Benjamin, 3 point problems then Bruno exchanges hats with Cindy (hence, after the second turn, Cindy has Ann s hat and Bruno has Cindy s). This goes on until each one have their own hat. How many exchanges are needed? (A) 4 (B) 6 (C) 8 (D) 12 (E) Sunday morning, the length of little worm Kesha was 1 cm. During the day his length has increased by half. And during the night his length has increased by one third compared to the evening value. How long was Kesha Monday morning? (A) 5/3 cm (B) 11/6 cm (C) 2 cm (D) 13/6 cm (E) 7/3 cm In a group of men and women, the average age is 31. If the men s average is 35 years and the women s average is 25, then the ratio of the number of men to the number of women is: (A) 5 : 7 (B) 7 : 5 (C) 2 : 1 (D) 4 : 3 (E) 3 : Mike has 1 euro. He decided to earn more money. He invested that 1 euro, and after one turn he had 2 euros. Then he saved 1 euro, and invested 1 euro again. After every turn his investment is doubled, and after every turn he saved 1 euro. How many euros he had in total after the 100th turn? (A) 99 (B) 100 (C) 101 (D) 102 (E) Tulio writes down the next list of natural numbers. Which is the last number that can be written by Tulio? 40, 42, 33, 35, 26, 28,... (A) 9 (B) 7 (C) 6 (D) 4 (E) 2 Benjamin Estela writes down a list of all integers that have three different digits, that she can form using the digits 0, 3, 4, 5 and 8. She puts the numbers in order, from minor to major. Which is the sum of the numbers that are in the 3 last places of the list? (A) 1033 (B) 971 (C) 917 (D) 1303 (E) Abel, Benito and Carlos compete with the jumping frogs, that they have constructed in a course of robotics, moving along a trail which is marked in centimeters. Abel s frog can make jumps of 60 cm, Benito s frog can make 80 cm in each jump and Carlos frog can make 15 cm in each jump. All of them start from the 0 cm mark. At what distance from the depart point meet all thre frogs for the first time? (A) 60 (B) 120 (C) 180 (D) 240 (E) Which of the following pieces covers the largest number of dots in the table? (Piece must lie completely on the table.) 45

46 Benjamin, 4 point problems (A) (B) (C) (D) (E) point problems In the equality W E + ARE = COOL the letters denote the digits of positive integers (different letters denote different digits). What is the biggest value that W + E can take?? (A) 14 (B) 15 (C) 16 (D) 17 (E) Find the sum of all the digits of (A) 7 (B) 2013 (C) 9314 (D) (E) other Benjamin ( )/2012 ( )/2012 = (A) 6 (B) (C) (D) 5 (E) All the three different digit numbers are formed using the distinct nonzero digits a, b, c. Find the sum of all these numbers divided by the sum of their digits. (A) 222 (B) 37 (C) a + b + c (D) a b c (E) Mary drawn figures on identical sheets of paper in form of square, one by one. How many of these figures have perimeter equal to the perimeter of the sheet of paper? (A) 2 (B) 3 (C) 4 (D) 5 (E) In 3/5 of the amount Tom received from his grandmother he bought a ticket for a football match. Left him another 14 dollars. How much has Tom received from his grandmother? (A) 21 USD (B) 35 USD (C) 50 USD (D) 42 USD (E) 70 USD How many od the chords designated by 10 points: A, B, C, D, E, F, G, H, I, J of the circle do not cut the diameter KL? 46

47 Benjamin, 4 point problems (A) 10 (B) 20 (C) 21 (D) 25 (E) The square in the picture, with each side of 1, is completely divided into squares. What is the value of the grey square area? (A) 1/2 4 (B) 1/2 5 (C) 1/2 6 (D) 1/2 7 (E) 1/2 8 Benjamin The following cards are placed so that a sum of three two-digit numbers is formed. What is the value of this sum? (A) 38 (B) 40 (C) 99 (D) 78 (E) To built a 5km fence, the support pillars should be placed each 50 metres. The pillars are square in shape with the side of 50cm. How many such pillars are needed? (A) 990 (B) 45 (C) 90 (D) 50 (E) There is five boxes in the store: 1 kg, 4 kg, 7 kg, 8 kg, 12 kg. If one of them will be sold, from the remaining 4 boxes it is possible to make two groups with the same weight. Which box will be sold? (A) 1 (B) 4 (C) 7 (D) 8 (E) Dan put a good addition of two 2-digit numbers on the magnetic board, but somebody make order after lession. On the board is seen the following: =. What was the 47

48 Benjamin, 4 point problems result of the addition? (A) 67 (B) 89 (C) 69 (D) 76 (E) On 23rd of March 2013 it is Saturday. What day will it be on 1st of June 2013? (A) Monday (B) Thursday (C) Friday (D) Saturday (E) Sunday A stack of stones must be transported. If Ann does the job alone, she needs one hour. If Ben does the job alone, he needs two hours. They decide to do the job together. What time will they need? (A) 30 min. (B) 40 min. (C) 1 hour (D) 1,5 hour (E) 3 hour Granddad used to walk every day. Nowadays he doesn t walk anymore on Wednesdays and Sundays. But on the other days he has extended his route with 40%. Is the total distance Granddad his walking in a week more or less than before? (A) The distance is 20% less (B) The distance is 10% less (C) The distance is the same (D) The distance is 10 % more (E) The distance is 20 % more Benjamin The picture shows how to make three squares using 10 matchsticks. How many matchsticks do you need to make 30 squares? (A) 66 (B) 71 (C) 75 (D) 77 (E) The train is 300 meters long. It takes the train 12 seconds to pass a semaphore. What is the speed of the train? (A) 90 km/h (B) 100 km/h (C) 120 km/h (D) 150 km/h (E) you can t know In a bottle is 10 liter water of a 6% salt content. This means 6 gram salt is dissolved in 1 liter water. Pete wants to make bottles of 100 ml water of 5% salt content of it. This means he has to dilute the original liquid. How many bottles of 100 ml can Pete make in this way of 10 liter? (A) 101 (B) 105 (C) 110 (D) 116 (E) Albert is making a bicycle tour of 60 km. He is riding with a constant speed of 18 km/h. Beatrice is making the same tour in the opposite direction with a constant speed of 22 km/h. Both start at 11:00. At what time to they meet? (A) 12:00 (B) 12:10 (C) 12:20 (D) 12:30 (E) 12:40 48

49 Benjamin, 4 point problems A shoe-lace is too long. The ends and the tags should be 1 cm shorter each. We will achieve this by shortening one end of the lace. With how many cm? (A) 1/6 cm (B) 1/3 cm (C) 1 cm (D) 3 cm (E) 6 cm Three brothers want to visit a fair. Unfortunately, their amount of money is insufficient. Albert has 3 euro, Bert has 5 euro and Chris 8 euro. Their father gives the three boys 20 euro together, after which each of the boys has the same amount of money. How much money does each boy have? (A) 5.33 EUR (B) 6.67 EUR (C) 12 EUR (D) 16 EUR (E) 20 EUR Benjamin John has made a building of cubes. In the picture you see this building from above. In each cell you see the number of cubes in that peculiar tower. When you look from the front, what do you see? (A) (B) (C) (D) (E) What is the remainder of 2013 : ( )? (A) 2 (B) 0 (C) 1 (D) 3 (E) some other number Father is 63 years old. His three sons are 35, 36 and 40 years old. How many years ago were the three sons together as old as their father? 49

50 Benjamin, 4 point problems (A) 8 (B) 16 (C) 24 (D) 32 (E) The diagram shows a clock face with 12 hour marks. How many different ways are there to draw a straight line through the clock face splitting the hour marks into two sets so that the hour numbers on each side of the line have the same sum? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 Benjamin How many triangles of any size are there in the diagram? (A) 12 (B) 18 (C) 20 (D) 30 (E) On Kangaroo Island there are seven lakes. 3 rivers flow from each lake and 2 rivers flow into them (each river flows from a lake and flows into a lake, or sea). How many rivers flow into the sea? (A) 2 (B) 4 (C) 6 (D) 7 (E) A 12-meter block has been sawed into three-meter logs within 12 minutes. How long does it take to saw the same block into one-meter logs? (A) 33 min. (B) 36 min. (C) 40 min. (D) 44 min. (E) 48 min What is the maximum number of figures that can be put in a cube 3 3 3? (A) 3 (B) 4 (C) 5 (D) 6 (E) How many two digit numbers have the 10 s digit larger than the units digit? (A) 36 (B) 45 (C) 54 (D) 90 (E) none of the previous Roo the kangaroo chose three numbers from the set 1, 2, 3, 4, 5, 6, 7, 8, 9. He then multiplied the three numbers. Which of the following could not be the result he found? (A) 27 (B) 35 (C) 39 (D) 64 (E) Matthew managed to shoot a few paper roses during the festival. His sister realized that if he shot three-times as many as he did, he would have 12 roses more. How many roses did Matthew shoot? (A) 7 (B) 6 (C) 5 (D) 4 (E) 3 50

51 Benjamin, 4 point problems Before his flight Michael had to weigh his empty bag. For measuring he used grandpa s old balance scale and 5 weights of 10 g, 30 g, 90 g, 300 g and 810 g. He found out that the bag s weight was 640 g. Which weight did he put to the same side of the scale as the bag? (A) 10 g (B) 30 g (C) 90 g (D) 300 g (E) 810 g Five small penguins from the Penguin family eat together as much fish per day as both parents together. Small penguins eat identical amount each, the parents eat identical amount each. The whole family eats 4 kg of fish per day. How many grams of fish does one small penguin eat on one day? (A) 2000 g (B) 1000 g (C) 800 g (D) 600 g (E) 400 g Gregory has as many brothers as sisters; however, his sister Mary has twice as many brothers as sisters. How many kids are in Gregory s family? (A) 4 (B) 5 (C) 6 (D) 7 (E) A gardener plants a certain number of trees in the first row. He plants twice this number in the second row, three times this number in the 3rd row and so on till he complete 9 rows of trees. If he wanted to rearrange these trees into a perfect square grid later on, how many should he plant in the first row? (A) 1 (B) 3 (C) 5 (D) 7 (E) In the squares of the 4 4 chessboard integers are written so that the numbers in adjacent squares differ by 1. We also know that numbers 3 and 9 are written there. How many different numbers are denoted in the table? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 Benjamin Inside a basket there are fruits. One third of the fruits are oranges, the rest are bananas and apples. The number of apples exceeds the number bananas in one half of the number of oranges. If there are 9 bananas in the basket, how many pieces of fruit are there altogether inside the basket? (A) 25 (B) 27 (C) 30 (D) 36 (E) If to buy at once n > 1 bars the same type of chocolate you should pay in total n EUR less than to buy n bars same chocolate separately. Andres bought 3 bars at once and paid 9 EUR. How much you should pay if you buy 5 bars of the same chocolate at once? (A) 13 EUR (B) 14 EUR (C) 15 EUR (D) 16 EUR (E) it is impossible to determine Mary was born on March 15th Today, on March 15th 2013 she wrote Find 100th number of the row. (A) 0 (B) 1 (C) 2 (D) 3 (E) The rectangular mirror was broken into seven pieces. Six of these pieces are shown in 51

52 Benjamin, 4 point problems figure on the right. Which of the following pieces is the 7th one? (A) (B) (C) (D) (E) Which of the following polygons cannot be a figure of intersection of a square and an equilateral triangle? (A) quadrangle (B) pentagon (C) hexagon (D) heptagon (E) all previous are possible Benjamin Among Nikolay s schoolmates there are twice more girls than boys. Which of the following numbers can be a number of all children which study in this class? (A) 30 (B) 20 (C) 24 (D) 25 (E) There are 8 kangaroes on the 4 4 board as shown in the figure. What least number of moves is necessary for exactly 2 kangaroes be in each row and in each column? Per one move one kangaroo can jump on next (by side or by vertex) empty cell. (A) 1 (B) 2 (C) 3 (D) 4 (E) Leo has a piece of butter that has the form of a cuboid. Before he makes a plane cut through the piece with a knife, he tries to figure out which of the following geometrical shapes are possible for the cutting surface: square; rectangle which is not a square; triangle; trapezium which is not a rectangle; pentagon. How many of these five shapes are possible? (A) 1 (B) 2 (C) 3 (D) 4 (E) There are 2 identical pencils and 2 identical rubbers. The weight of 2 pencils together with one rubber is 20 g but the weight of one pencil together with 2 rubbers is 25 g. How much is the weight of one pencil? (A) 3 g (B) 5 g (C) 6 g (D) 8 g (E) 10 g The final of the local hockey championship was a match full of goals. There were 6 goals in the first half and the guest team was leading after the first half. After the home team 52

53 Benjamin, 4 point problems shot 3 goals in the second half, they won the game. How many goals did the home team shoot altogether? (A) 3 (B) 4 (C) 5 (D) 6 (E) Yesterday, when I found myself face to face with the Kangaroo, the average of the times left to noon and left to midnight was giving the exact time of our encounter. What time was it then? (A) 5 : 00 (B) 6:00 (C) 8:00 (D) 9:00 (E) 10: The perimeter of a square is 2012cm. What is its area? (A) cm 2 (B) cm 2 (C) cm 2 (D) cm 2 (E) cm How many of the addition symbols (+) in the incorrect statement = 27 must be changed to multiplication symbols ( ) to make the statement true? (A) 1 (B) 2 (C) 3 (D) 4 (E) it is impossible Nick learns to drive. For the moment, he only can turn to the right and cannot turn to the left. What is the smallest number of turns he must do to get from A to B? Benjamin (A) 3 (B) 4 (C) 6 (D) 8 (E) What is the last digit of the smallest 3-digit number divisible by 6 and having exactly two different digits in its decimal notation? (A) 0 (B) 1 (C) 2 (D) 4 (E) How much time will pass from the middle of the first half of an hour till the middle of its third quarter? (A) 22 min 30 sec (B) 15 min (C) 15 min 30 sec (D) 20 min 15 sec (E) 25 min On his birthday, grandfather said: Today I ve started my eighth decade. His grandson, who likes to count in dozens, remarked: Grandpa, you are in your... dozen. What word is missing? (A) fifth (B) sixth (C) seventh (D) eighth (E) ninth Mike has 1 euro. He decided to earn more money. He invested that 1 euro, and after one turn he had 2 euros. Then he saved 1 euro, and invested 1 euro again. After every turn his 53

54 Benjamin, 5 point problems investment is doubled, and after every turn he saved 1 euro. Which of the following expression can describe how many euros will he have after the nth turn? (A) n (B) n + 1 (C) 2n (D) 2n + 1 (E) 2n Alex counts the letters of his name the right way and back again, over and over, as indicated. Number 13 is in this way given to the letter A. Below the letter L you can see among the numbers all the multiples of 6: 6, 12, 18,..., etc. Bardan is going to count the letters of his name in the same way. Of which number you can see all its multiples below the same letter? (A) 2 (B) 4 (C) 5 (D) 8 (E) 10 Benjamin Aron, Bern and Carl always lie. Each of then owns a red stone or a green stone. Aron says: My stone is of the same color than Bern s stone, Bern says: My stone is of the same color than Carl s stone. Carl says: Exactly two of us own red stones. Which of the following asertions is true? (A) Aron s stone is green (B) Bern s stone is green (C) Carl s stone is red (D) Aron s and Carl s stones are of different color (E) none of them From a list of three numbers we call changesum the procedure to made a new list replacing each number by the sum of the other two. For example, from {3, 4, 6} changesum gives {10, 9, 7} and a new changesum leads to {16, 17, 19}. If we begin with the list {20, 1, 3}, what is the maximum difference between two numbers of the list after 2013 consecutive changesums? (A) 1 (B) 2 (C) 17 (D) 19 (E) We fill a rectangular grid of 2013 squares with all the integers from 1 to 2013, each number once. We write it in increasing order, filling first row 1, after row 2, etc.. If the number 50 is in the second row and number 100 in the 4th row, in which row will be located number 1000? (A) 39 (B) 40 (C) 30 (D) 31 (E) point problems All 4-digit positive integers with the same four digits as in the number 2013 are written in one line without gaps and in an increasing order. Which of the following combinations of two digits appears less times in this sequence of digits than the other four? (A) 01 (B) 30 (C) 02 (D) 12 (E) 31 54

55 Benjamin, 5 point problems The sum of the following Roman numbers: M - D, MC - D, MCC - D, MCCC - D, MCD - D, MD - D, out of which you substract the Roman number MM is: (A) MMD (B) MM (C) MMM (D) M (E) D Andrei (A) is going to the swimming pool (P). His mother says: You are allowed to walk only from the left to the right and up to down in the streets on the sketch. In how many ways can Andrei get to the pool? (A) 4 (B) 6 (C) 1 (D) 3 (E) Find the number of pairs of two-digit natural numbers whose difference is equal to 50. (A) 40 (B) 30 (C) 50 (D) 60 (E) Kangoo Lucky loves travelling through tunnels by train. Yesterday he measured the time when he entered the tunnel (12:30) and when he came out again (12:34). How long was Lucky in the tunnel? (A) exactly four minutes (B) at most four minutes (C) at least four minutes (D) at least three minutes (E) more than four minutes How many two-digit numbers have the property that the product of the two digits is an odd number? (A) 50 (B) 25 (C) 40 (D) 20 (E) 16 Benjamin (A) (B) (C) (D) (E) Put one of these numbers 0, 1, 2, 3, 5, 6, 7 in each box so that the operation is correct. What is the value of this operation result? (A) 120 (B) 123 (C) 176 (D) 167 (E) Put one of these numbers 0, 1, 3, 4, 5, 6, 7, 8, 9 in each box so that the operation is correct. What is the value of this operation result? 55

56 Benjamin, 5 point problems (A) 863 (B) 134 (C) 865 (D) 765 (E) In the sum = 1413 the middle digits are invisible. What is the larger of the two invisible numbers? (A) 0 (B) 1 (C) 2 (D) 3 (E) None of the previous Ann drives on her bicycle with constant speed. In the picture you see her watch in the beginning and at the end. Which picture shows the position of the minutes-arm when Ann finishes one third of the ride? (A) (B) (C) (D) (E) Benjamin Alex like to place five stones on the 5 5-grid, in such a way that there are no two stones in the same row, column or diagonal. Alex places the first stone in the central cell. In how many ways he can finish the job? (A) 0 (B) 1 (C) 2 (D) 3 (E) A large cube has been built with little cubes of five different patterns, in a regular way. What is the central little cube? 56

57 Benjamin, 5 point problems (A) (B) (C) (D) (E) The water level of the sea reaches its maximum (flood) every 12 hours and 24 minutes. Today, at 12:00, it was flood-tide. How many days will it last before it is flood-tide at 12:00 again? (A) 10 (B) 30 (C) 31 (D) 60 (E) Mark plans to visit a friend who lives 2 Km from him. There is a straight street connecting their houses. Mark plans to bring his dog with him and wants the dog running as much as possible, so when walking throws a ball ahead and orders the dog to pick the ball and to bring it back to him, and so on. Marks walks at 4 Km per hour, his dog can run at 18 Km per hour. What is the maximum length in Km the dog can cover playing in this way? (A) 2 (B) 4 (C) 5 (D) 8 (E) A different number boys and 28 girls stay in a circle, hand by hand. Exactly 18 boys give their right hand to a girl. How many boys give their left hand to a girl? (A) 18 (B) 9 (C) 28 (D) 14 (E) The diagram shows an incomplete magic square that uses the first nine odd numbers. Benjamin What number goes in the central cell? (A) 7 (B) 9 (C) 11 (D) 13 (E) One of two kangaroo friends, Kan and Roo, stole a candy. When the teacher asked them what happened - Kan said I stole the candy, - Roo said I did not steal the candy. If at least one of the two lied, which of the following is true? (A) Kan stole the candy (C) none of the two stole the candy (E) both Kan and Roo said the truth (B) Roo stole the candy (D) Kan said the truth and Roo lied Mary wrote down a number in every square of the strip so that the sum of the numbers in any three consecutive squares was the same (see fig.), and then she counted the sum of all the squares of the strip. The sum appeared to equal 20. Some numbers blotted. What number was written in the second square? (A) 2 (B) 3 (C) 4 (D) 6 (E) There are 11 boys more than girls among Ivanko s classmates. The boys in the class are three times as many as the girls. Irynka is Ivanko s classmate. How many girls are Irynka s 57

58 Benjamin, 5 point problems classmates? (A) 5 (B) 6 (C) 7 (D) 12 (E) such situation is impossible Volodya and Pavlyk collect stamps. If Volodya gave Pavlyk as many stamps as he has got now, both of the boys would have the equal number of them. Volodya gave Pavlyk some stamps (no more than five), and the rest put equally in three albums. How many stamps did Volodya give to Pavlyk? (A) 1 (B) 2 (C) 3 (D) 4 (E) Jacob and Will were making big cubes from small wooden ones. They both used 27 cubes each. Jacob painted all sides of the cubes and than glued them together into a big cube. Will did it the other way. He glued the big cube first and than painted it red on all sides. How many times more color did Jacob use compared to Will? (A) Two times. (B) Three times. (C) Four times. (D) Five times. (E) Six times. Benjamin When Pinocchio lies, his nose gets 8 cm longer. When he says the truth, the nose gets 3 cm shorter. When his nose was 7 cm long, he said five sentences and his nose got 25 cm long after that. How many of Pinocchio s sentences were true? (A) 1 (B) 2 (C) 3 (D) 4 (E) Kevin and Jim played poker with 1-cent and 2-cent coins. At the beginning they both had 10 cents each, Jim having 5-times more 1-cent coins than Kevin. How many coins did the boys play with? (A) 12 (B) 14 (C) 16 (D) 18 (E) cats signed up for the contest MISS CAT After the first round, 21 got out, because they failed in catching mice. 27 cats out of those that remained in the contest had stripes and 32 of them had one black ear. All striped cats with black ear got to the final. What is the minimum number of the finalists? (A) 5 (B) 7 (C) 13 (D) 14 (E) regular dice are clued together to from a cube. What is the maximum number of dice with six pips that can be seen on the outside of the cube? (A) 27 (B) 26 (C) 24 (D) 21 (E) Father Kangaroo writes down digits 1 to 5 and asks his daughter kangaroo to form a random 4 digit number without repeating any digit. What is the probability that the number formed by daughter kangaroo is divisible by 12? (A) 1/12 (B) 3/16 (C) 23/120 (D) 1/20 (E) 7/ A bike rider plans to ride 10 km in exactly 15 minutes. To do this he must keep an average speed of 40 km/h. After 9 km is he right on schedule. Average speed so far is exactly 40 km/h. The final kilometer consists of an uphill of 500 m and a corresponding downhill. Up the hill he bikes with an average speed of 20 km/h. 58

59 Benjamin, 5 point problems How fast must he ride the last 500m towards targets to clear time for exactly 15 min.? (A) 40 km/h (B) 60 km/h (C) 80 km/h (D) 100 km/h (E) It is impossible In the squares of the 4 4 chessboard integers are written so that the numbers in adjacent squares differ by 1. We also know that numbers 3 and 9 are written there. Find the sum of all the numbers written in the table. (A) 96 (B) 72 (C) 48 (D) 40 (E) Both Grandmother and her granddaughter were born on 21 of March. This year, on their common birthday the number of Grandmother s years will be equal to the number of granddaughter s months and the sum of their ages will be 78. By how many years Grandmother is older than her granddaughter? (A) 56 (B) 60 (C) 66 (D) 72 (E) How many 3-digits numbers possess the following property: subtracting 297 from such a number, we get a 3-digit number written with the same digits in the reverse order? (A) 6 (B) 7 (C) 10 (D) 60 (E) There were 2013 inhabitants on an island, some of them knights and the other liers. The knights only say truth and the liars always lie. Every day one of the inhabitants said: After my departure the number of knights on the island will equal the number of liars and then left the island. After 2013 days there was nobody on the island. How many liars were there initially? (A) 0 (B) 1006 (C) 1007 (D) 2013 (E) impossible to determine Benjamin Three painters should paint 10 identical 4-faced pyramids. Each side of a pyramid should be painted completely only by one painter. Different painters can paint different sides of the same pyramids, but no two of them can paint the same pyramid simultaneously. For painting of any side of a pyramid 10 seconds are required to each painter. Painters begin to work simultaneously. For what least time painters can paint completely all 10 pyramids? (A) 130 seconds (B) 140 seconds (C) 150 seconds (D) 160 seconds (E) 170 seconds There is playground on the diagram. Getting on a shaded circle you should move back on a circle indicated by an arrow. In the beginning the button is on START. Alex rolled the die 6 times and his results were 1, 2, 3, 4, 5, 6 in some order, and he reached to the FINISH. How many pips should he have had on the first rolling? (A) only 2 or 5 (B) only 1 or 3 (C) only 2 or 3 (D) only 4 (E) only There are four buttons on the line. Two of them have happy face, and two of them have sad face. If to press on a face, its expression turns to the contrary (e.g. a funny face turns into a sad face after the touch). In addition to that, the bordering pictures of the pressed button change expressions opposite also. Following the diagram, how many times at least should you press on the buttons in order to get all happy faces on the line? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 59

60 Benjamin, 5 point problems Construct a cube by 4 white and 4 black unit cubes. How many different cubes could be constructed in this way? (Two cubes are not different if the one could be obtained by rotating the other.) (A) 16 (B) 9 (C) 8 (D) 7 (E) Using the digits 1, 5 and 9 Ben made three various three-digit numbers (in each number each of these digits used exactly once). To what of the following values the sum of these three numbers cannot be equal? (A) 1665 (B) 1953 (C) 945 (D) 2385 (E) The rectangle consists of six squares as shown. Find the perimeter of the rectangle if its length is equal to 546 cm. (A) 2006 (B) 2016 (C) 2026 (D) 2036 (E) 2046 Benjamin How many neighbour couples of months with an odd number of days are there in the period ? (A) 7 (B) 11 (C) 13 (D) 15 (E) The perimeter of a triangle with the sides a, b and c is equal to 90 cm. It is known, that side b is 50 % shorter than side a, and the side c is 50 % longer than side b. Find the length of the least side of this triangle. (A) 10 cm (B) 15 cm (C) 20 cm (D) 25 cm (E) 30 cm children (boys and girls) stand in a line. The number of children between any two boys does not equal 2. What the greatest number of boys can be among these 10 children? (A) 4 (B) 5 (C) 6 (D) 7 (E) the situation is not possible Winnie Pukh left his house (marked V on the scheme) in the morning. For a day he visited all his friends and remained to spend the night in the house of Donkey Ia-Ia. Winnie Pukh visited some of friends twice, but each path on the scheme he passed exactly once. In 60

61 Benjamin, 5 point problems what house Ia-Ia lives? (A) 1 (B) 3 (C) 5 (D) 7 (E) impossible to determine February 20th, March 12th, March 20th, April 12th and April 23th are birthdays of Andy, Betty, Cathie, Dannie and Eddy. Andy and Eddy were born in the same month, Betty and Cathie were born in the same month as well. Andy and Cathie were born in the same day number of different months, Danny and Eddy were born in the same day number of different months as well. What day is Betty s birthday? (A) February 20th(B) March 12th (C) March 20th (D) April 12th (E) April 23th Mandy wants to pick flowers in the park. There are beds arranged in a long row: tulips, daffodils, forget-me-nots, then again tulips, daffodils, forget-me-nots,... In which of the following ways can she pick the flowers to get a bouquet of four flowers of each sort? (A) Take a flower from every third bed. (B) Don t take a flower from every sixth bed. (C) Take in turns two flowers and leave one flower. (D) Take in turns four flowers and leave two flowers. (E) Take in turns four flowers and leave one flower. Benjamin When Matthew and Marten found their old model railway, Matthew quickly made a perfect circle from 8 identical track parts. With the same type of track parts Marten tries a little more difficult track starting with 2 of these pieces. He wants to use as few pieces as possible for his closed track. How many pieces does his track consist of? (A) 11 (B) 12 (C) 14 (D) 15 (E) Charley wants to exchange the 3 white tokens with the 3 black tokens and can do it by moves. A move is either to push a token to an empty neighbouring square or to jump over one token in case the following square is empty. What smallest number of moves is necessary? (A) 15 (B) 16 (C) 17 (D) 18 (E) Four strips of 3, 5, 11 and 13 cm long are fixed with one end common as shown in the figure. The other ends of the strips are the vertices of a quadrilateral. What is the maximum 61

62 Benjamin, 5 point problems area that can have this quadrilateral? (A) 142 cm 2 (B) 98 cm 2 (C) 96 cm 2 (D) 128 cm 2 (E) 126 cm Alix was 22 last week. When she adds together her age and the last two digits of the year she was born (that is, ignoring the nineteen hundreds ), she gets = 113. Alix s elder sister Beatrix also had a birthday last week. When Beatrix does the same for her corresponding numbers, what answer should she get? (A) 110 (B) 111 (C) 112 (D) 113 (E) More information is needed Benjamin Pinocchio arranges the integers 1, 2, 3,..., 100 around a circle. Father Carlo promised him a golden coin for each number that is greater than the sum of its neighbors. What is the largest number of golden coins Pinocchio can get? (A) 99 (B) 50 (C) 49 (D) 25 (E) N is a natural number. Among (A) - (E), what number can be the difference between the sum of digits of N and that of N + 1? (A) 2011 (B) 2012 (C) 2013 (D) 2014 (E) From 4 identical cubes (the net of such a cube is shown in Figure 1), Alice glues a bar shown in Figure 2 (only faces with identical numbers are glued together). What is the largest sum that Alice can get on the bar surface? (A) 66 (B) 68 (C) 72 (D) 74 (E) Two cats Tony and Tiny are napping on a sofa. Tony went to nap first, and then Tiny took a quarter of the free surface. Together, they occupied exactly half of the sofa. What part of the sofa was occupied by Tiny? (A) 1 2 (B) 1 5 (C) 1 6 (D) 1 8 (E) What is the smallest possible number of elements in a set of natural numbers with exactly 4 numbers divisible by 2, exactly 3 numbers divisible by 3, and exactly 2 numbers divisible by 6? (A) 4 (B) 5 (C) 6 (D) 10 (E) Basil is encoding numbers as follows: starting with an integer, he writes down the product of its first two digits, then he multiplies the second and the third digits, and so on. 62

63 Benjamin, 5 point problems For instance, the number 5648 will become How many numbers can turn to 5648? (A) 0 (B) 1 (C) 2 (D) 3 (E) Among n natural numbers 1, 2, 3,..., n, at least 13 are divisible by 4 and at most 9 are divisible by 6. How many of these numbers are divisible by 12? (A) 3 (B) 4 (C) 5 (D) 6 (E) impossible to determine What is the smallest number of consecutive integers such that exactly 5 of them are divisible by 5 and exactly 7 of them end with 7? (A) 21 (B) 22 (C) 23 (D) 25 (E) Basil chose several numbers among the integers 1, 2, 3,..., 30. It turned out that exactly 4 of them are divisible by 4, exactly 3 are divisible by 6, and exactly 4 are divisible by 5. What is the smallest possible number of integers in Basil s collection? (A) 4 (B) 6 (C) 7 (D) 9 (E) Let n N 0. The number of integers that are less than n and greater than n equals: (A) n 1 (B) 2n 1 (C) 2n (D) 2n + 1 (E) infinitely many Alex counts the letters of his name the right way and back again, over and over, as indicated. Number 13 is in this way given to the letter A. Below the letter L you can see among the numbers all the multiples of 6: 6, 12, 18,..., etc. In the same way you can see all the multiples of some other numbers. Martin is going to count the letters of his name in the same way. In his name you can also see all the multiples of some numbers. What formula gives you the smallest such number S from the number of letters N of the name? Benjamin (A) S = 1.5N (B) S = N + 2 (C) S = 2N 2 (D) S + N = 10 (E) Such a formula does not exist In the equality (a + b c) (d + e f) = 2013 the numbers a, b, c, d, e, f are the digits 4, 5, 6, 7, 8, 9 in some order. So each of ten digits, 0 through 9 appears once in the given equality. If a < b < c < d what is the value of e f? (A) 24 (B) 72 (C) 30 (D) 42 (E) It may have different values Admission fees in the auditorium of Science Park are 75 cents for adults and 25 cents for young people. One afternoon in the auditorium of the Park, with a capacity for 600 people and it was not full, was collected cents. How many adults attended at least? (A) 359 (B) 300 (C) 365 (D) 361 (E)

64 Benjamin, 5 point problems We color the cells in a rectangular grid alternately black and white like a chessboard. If we made this procedure on a grid composed exactly by 2013 cells, and we start painting white one of the four corners of the rectangle, how many cells will be coloured white? (A) 1006 (B) 1007 (C) 976 (D) 1037 (E) It depends on the dimensions of the rectangle Benjamin 64

65 Cadet, 3 point problems 4 Cadet point problems The number 2013 has the property that its last digit equals the sum of the first three. How many year numbers have this feature in our millennium (starting in 2001 and ending in 3000)? (A) 24 (B) 36 (C) 27 (D) 64 (E) In the equality W E + ARE = COOL the letters denote the digits of positive integers (different letters denote different digits). What is the smallest value that W E can take? (A) 2 (B) 3 (C) 6 (D) 8 (E) Let p and q be two prime numbers such that p 2 + q 2 = 365. Then p + q is: (A) 20 (B) 21 (C) 22 (D) 23 (E) other Let p and q be two prime numbers such that p 4 + q = Then p + q is: (A) 45 (B) 107 (C) 981 (D) 1999 (E) other If x = then x x x x is equal to: (A) 1 (B) 0 (C) 1 (D) (E) other How many solutions (x, y), where x and y are non-negative integers, does the equation y + 3x = 2013 have? (A) 1 (B) 671 (C) 672 (D) 673 (E) If a 6 = b and b 2 = c 3, where a, b, c are natural numbers, what relation is correct? (A) c = a (B) c = a 2 (C) c = a 4 (D) c = a 6 (E) c = a 12 Cadet The result of the following calculation is 2012: How many substractions have been done in this calculation? (A) 2012 (B) 1005 (C) 1006 (D) 503 (E) How many zeros are there at the end of the number ? (A) 0 (B) 1 (C) 2 (D) 4 (E) If a + b + c = 2 2 3, what is the value of the sum of following two-digit numbers: ab, bc, ca (the numbers are formed using the distinct nonzero digits a, b, c)? (A) 132 (B) 48 (C) 72 (D) 51 (E) Calculate: 1/2 + (1/2 + 1/3) + (2/3 + 1/4) + (3/4 + 1/5) + (4/5 + 1/6) + (5/6 + 1/7) + (6/7 + 1/8) + (7/8 + 1/9) + (8/9 + 1/10) (A) 7.1 (B) 8 (C) 8.5 (D) 8.1 (E)

66 Cadet, 3 point problems In London 2012 in 100 m freestyle swimming the winner was Nathan Adrian with 47,52 sec, on the second place James Magnussen with 47,53 sec. What was the estimate different in finish beetween them? (A) 2 mm (B) 2 cm (C) 2 dm (D) 2 m (E) 20 m Calculate the value of ( 1 + ) (A) 2017 (B) 2017 (C) 2017 (D) (E) None of the previous Which of the following calculations is incorrect? (A) = 1000 (B) = 1000 (C) = 1000 (D) = 1000 (E) = A heavy southern wind is blowing. Therefore an airplane is flying from Amsterdam to Paris with a velocity of 700 km/h. Another airplane is at the same time flying from Paris to Amsterdam with a velocity of 900 km/h. The flight Amsterdam-Paris takes 36 minutes. How long does the flight Paris-Amsterdam take? (A) 18 min. (B) 20 min. (C) 25 min. (D) 28 min (E) 36 min a and b are two digits. Their product ab ba is a number of four digits. It ends on 3, so ab ba = 3. What is the first digit of the product? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 Cadet Lisa has six pieces like the one in the picture. She tries to fit them in the figure with 24 squares. The first piece Lisa has placed already. At how many ways she can finish the job? (A) 2 (B) 3 (C) 4 (D) 5 (E) ,13 : 0,13 = (A) 101 (B) 111 (C) 1010 (D) 1111 (E)

67 Cadet, 3 point problems If a footpath makes a turning, you ll need special pieces. Such a piece makes an angle of 135. How much of these pieces you need to make a closed circuit? (A) 6 (B) 8 (C) 9 (D) 10 (E) The average number of children in five families cannot be: (A) 0.2 (B) 1.2 (C) 2.2 (D) 2.4 (E) tennis players are playing a tournament. In the first round 16 of them play against the 16 others. The losers are eliminated. In the second round 8 winners of the first round play against the other 8 winners. The winners of the second round play the quarter finals, the winners of those quarter finals play the semi finals and finally the final is played. What is the average number of games per player? (A) 31/32 (B) 1 (C) 1 1/2 (D) 31/16 (E) is divisible by 11 and is made of the four digits 0, 1, 2 and 3. What is the smallest such number? (A) 1023 (B) 1032 (C) 1203 (D) 1230 (E) 2013 Cadet is divisible by 11 and is made of the four consecutive digits 0, 1, 2 and 3. What is the smallest next such number? (A) 2031 (B) 2103 (C) 2130 (D) 2301 (E) Mark and Liza run around a circular fountain. At the beginning they are at the opposite ends of a diameter of the fountain, then they start running in the same direction. If Mark s speed is 9/8 of Liza s, how many loops has Liza completed when Mark reaches her? (A) 4 (B) 8 (C) 9 (D) 2 (E) It depends on the diameter of the fountain white and 18 black cubes with edge 1 are used to form one big cube with edge 3. The partial area of the new cube that is white, is at most: 67

68 Cadet, 3 point problems (A) 1/2 (B) 13/27 (C) 25/54 (D) 4/5 (E) 1/ The diagram shows a rhombus made up of 8 equilateral triangles. What is the sum of the total number of triangles in the figure and the number of rhombuses similar to the largest rhombus? (A) 9 (B) 11 (C) 18 (D) 19 (E) The kangaroo knew that 1111 = 11. Use this to help him calculate quickly the value of (A) 9999 (B) (C) 5 (D) 55 (E) none of the previous Which of the following numbers is the largest? (A) ( 1) 2013 (B) ( 1) ( 1) 2013 (C) ( 3) ( 3) 2013 (D) ( 2) ( 2) 2013 (E) ( 1) ( 1) ( 1) The ratio of a head size to human height depends on the age of a human. It can be estimated approximately using the equation H = 9 : (2a + 36), where H is the head size and a is the age of the child. What part of the total height does a head of a 9-year old child take? (A) 1/4 (B) 1/5 (C) 1/6 (D) 1/7 (E) 1/8 Cadet Flat rectangular roof, 11 m long and 6 m wide, got covered with 25 cm thick even layer of snow. How many m 3 of snow fell on the roof? (A) 8,25 m 3 (B) 16,5 m 3 (C) 66 m 3 (D) 825 m 3 (E) 1650 m The sea water contains the amount of salt and water in the ratio 7 : 193. How many kilograms of salt is there in 1000 kg of sea water? (A) 35 kg (B) 186 kg (C) 193 kg (D) 200 kg (E) 350 kg In the triangle ABC, let O be the point where the angle bisectors AA 1 and BB 1 intersect. It is known that AOB = 5 ACB. Find the angle ACB. (A) 10 (B) 20 (C) 30 (D) 36 (E) Martin has a cube with edges of length 10 cm. He wants to divide it into unit cubes and make smaller cubes from them, each of them built with more than one cube and all of them having different edges. What is the maximal number of cubes that Martin can build? (A) 2 (B) 3 (C) 4 (D) 5 (E) Look at the sequence of images. 68

69 Cadet, 3 point problems Which images completes the previous sequence? (A) (B) (C) (D) (E) Let C 1 and C 2 be two different circles with centre O. Let [AB] be a chord of the C 1 that is tangent to C 2 and such that AB = 20cm. What is the area of the grey region? (A) 2πcm 2 (B) 10πcm 2 (C) 20πcm 2 (D) 100πcm 2 (E) 120πcm All the students from the 9th grade from College Kangaroo attended an exam in room B.1 at 9 o clock. At 10 o clock, 15 girls left the room and the number of boys remaining in the room was twice the number of girls remaining in the room. At 11 o clock, 31 boys left the room, and the number of remaining boys and remaining girls in the room became the same. The total number of students from the 9th grade from College Kangaroo is (A) 33 (B) 50 (C) 80 (D) 100 (E) In country X, vehicle registration plates are composed by two digits followed by 4 letters; for instance 12 MAST. The first number in a plate cannot be zero. How many of such registration plates exist with numbers and letters taken from MATHS 2013? (A) 20 (B) 33 (C) 1440 (D) 7500 (E) Cadet If α = 35, β = 40 and γ = 55 then δ = (A) 100 (B) 105 (C) 120 (D) 125 (E) What is the sum of the interior angles of a triangle plus the sum of the interior angles of a quadrilateral minus the sum of the interior angles of a pentagon? (A) 0 (B) 45 (C) 90 (D) 135 (E) x, y and z are positive whole numbers; x y = 14,y z = 10 and z x = 35. What is the value of x + y + z? 69

70 Cadet, 3 point problems (A) 10 (B) 12 (C) 14 (D) 59 (E) Mary reads quickly but her daughter reads far more quickly. Mary s daughter reads twice as many words as Mary in one third of the time. Mary can read 120 words in a minute. How many words can her daughter read in one minute? (A) 240 (B) 320 (C) 480 (D) 720 (E) What is the value of ? (A) 6 (B) 45 (C) 51 (D) 65 (E) Which of the following is not equal to the product of the sum of its digits and a positive integer? (A) 512 (B) 444 (C) 410 (D) 332 (E) Ruy had 10c and 20c coins. He had more 10c coins than 20c coins and the total number of coins was smaller than 14. He used all his coins to buy a cake and he forgot how much he paid. However, he remembers that when he tried to put the coins of each kind on piles of size 2, one of each kind was left out and when he tried to make piles of three coins of each kind, 2 of each kind were left out. How much did he pay? (A) 1.5 euros (B) 1.8 euros (C) 2 euros (D) 2.1 euros (E) cannot say In the picture, the big triangle is equilateral and has area 1. The lines are parallel to the sides and divide the sides into three equal parts. What is the area of the shaded part? Cadet (A) 1 3 (B) 1 2 (C) 3 2 (D) 2 3 (E) Masha created a new algebraic operation: a b = a + 2b, and then calculated (a a) (b a). What was her result? (A) 7a + 2b (B) 5a + 2b (C) 3a + b (D) 5a + b (E) 7a + b For the number 2013, the squared sum of the cubes of its digits is (A) 6 4 (B) 6 6 (C) (D) (E) Following a pattern, Tulio writes down a list of numbers. Which is the last natural number that can be written by Tulio? 40, 42, 33, 35, 26, 28,... (A) 9 (B) 6 (C) 5 (D) 4 (E) Cheli is looking for integers between 200 and 300, such that by summing its three digits, she obtains the same result as if she multiplies these three digits. Which is the sum of 70

71 Cadet, 3 point problems the numbers found by Cheli? (A) 340 (B) 366 (C) 440 (D) 444 (E) Carmen should calculate the sum of all the divisors of the number 18. Which is the result obtained by Carmen? (A) 42 (B) 41 (C) 39 (D) 21 (E) There is a bag with 2 red balls, 3 blue balls, 10 white balls, 4 green balls and 3 black balls. Every student of the class extracts a ball of the bag without looking. What is the least number of children that should extracts balls of the bag, to be sure to have two balls of the same color? (A) 2 (B) 12 (C) 10 (D) 5 (E) In a meeting of parents of the sixth grade, they chose a deputy. Five candidates were proposed. The candidate who won had got 10 votes and there were not two candidates with the same number of votes. Every candidate obtained the greater number of possible votes. Which is the number of parents who were present in the meeting? (A) 34 (B) 40 (C) 30 (D) 36 (E) different points are marked in two parallel lines p and q. 6 points are marked on p and 4 on q. Then all the possible segments, with edges on p and q, are drawn. Find the difference between the segments with edges on p and segments with edges on q. (A) 0 (B) 2 (C) 3 (D) 4 (E) On the plane, we have three points A, B and C that form a triangle. With the triangle as a base, we want to draw a new parallelogram, by adding a fourth point as a vertex. How many different points on the plane can be chosen? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 Cadet A stack of boxes of 20 g, 30 g and 40 g is shown in the figure. Dylan stacks weights of the same type and then weighs them all, obtaining 380 g. Which is the quantity of weights of 30 g there cannot be? (A) 2 (B) 3 (C) 4 (D) 6 (E) Vero, Cheli and Pao draw the polygons that can be seen in the figure. They paint their polygons as quick as the others. The square of Vero has 9 cm of side, the rectangle of Cheli 12 by 8 cm and the rhombus of Pao has 71

72 Cadet, 4 point problems diagonals of 10 cm and 16 cm. In which order do they finish the polygons painting? (A) Vero, Cheli, Pao (B) Vero, Pao, Cheli (C) Cheli, Pao, Vero (D) Pao, Vero, Cheli (E) Pao, Cheli, Vero Which of the following pieces covers the largest number of dots in the table? (Piece must lie completely on the table.) (A) (B) (C) (D) (E) point problems All 4-digit positive integers with the same four digits as in the number 2013 are written on the blackboard in an increasing order. What is the largest possible difference between two neighbouring numbers on the blackboard? (A) 702 (B) 703 (C) 693 (D) 793 (E) For a 3-digit number ABC with A > B > C, what is the largest possible sum of digits of the number 9 ABC? (A) 9 (B) 18 (C) 27 (D) 36 (E) Let S(n) denote the sum of the digits of the integer n. Then the alternating sum Cadet S(1) S(2) + S(3) S(4) + + S(2011) S(2012) + S(2013) evaluates to: (A) 0 (B) 1006 (C) 1007 (D) 2013 (E) other Find the probability of choosing a random card from a standard deck that is a club. (A) 13/54 (B) 1/4 (C) 12/54 (D) 1/2 (E) 23/ Franois says Each of my children has as many children as siblings (sisters and brothers) ; and my age is the number of all my children and grandchildren. Knowing that Franois is between 60 and 75, how old is he? (A) 62 (B) 64 (C) 67 (D) 70 (E) A firm increases the area of its rectangular tablets by increasing the length by 80% and reducing the width by 20%. By what percentage do it have to increase the side of its square tablet so that its area varies in the same proportions as the rectangular ones? (A) 50% (B) 44% (C) 40% (D) 30% (E) 20% 72

73 Cadet, 4 point problems The cube root of 3 33 is equal to (A) 3 3 (B) (C) 3 23 (D) 3 32 (E) ( 3) divided by is equal to (A) (B) (C) (D) 2 15 (E) The perimeters of the triangles ADF, DBE, DEF and F EC are 12, 24, 19 and 24 as shown. What is the perimeter of the triangle ABC? (A) 38 (B) 41 (C) 43 (D) 47 (E) In the cube below you see a pyramid ABCDS with base ABCD, whose vertex S lies exactly in the middle of the corresponding edge of the cube. You look at this pyramid from above, from below, from behind, from ahead, from the right and from the left. Which view does not arise? Cadet (A) (B) (C) (D) (E) Triangle RZT is the image of the equilateral triangle AZC upon rotation around Z, whereby β = CZR = 70. Determine the angle α = CAR. 73

74 Cadet, 4 point problems (A) 20 (B) 25 (C) 30 (D) 35 (E) How many two-digit even numbers have the property that the sum of the two digits is also an even number? (A) 50 (B) 25 (C) 40 (D) 20 (E) Determine the sum of the digits of the smallest possible number with 24 as the product of its digits. (A) 6 (B) 8 (C) 9 (D) 10 (E) In equilateral triangle ABC we know that AE = BD = AB/3. What is the measure of angle DF C? (A) 10 (B) 15 (C) 30 (D) 145 (E) 60 Cadet Any point D, which is interior to the triangle ABC, determines vertex angles having the following measures: m(bad) = x, m(abd) = 2x, m(bcd) = 3x, m(acd) = 4x, m(dbc) = 5x. Find the measure of x. (A) 5 (B) 6 (C) 9 (D) 10 (E) How many three-digit numbers have the property that the sum of the digits is 10? (A) 10 (B) 20 (C) 54 (D) 81 (E) Mihaela wrote on the chalkboard the bottom of the numbers triangle. Ioana begins to fill the top rows as follows: on the superior row, between any two numbers on the inferior row, there lies the sum of these two numbers. What number will be written on the top of the triangle? 74

75 Cadet, 4 point problems (A) 2012 (B) 2816 (C) 2826 (D) (E) Sara wrote on the chalkboard the bottom of the numbers triangle. Maria begins to fill the top rows as follows: on the superior row, between any two numbers on the inferior row, there lies the product of these two numbers. What number will be written on the top of the triangle? (A) (B) (C) (D) (E) Five consecutive positive integers have the following property: there is three of them with the same sum than the sum of other two. E.g. 2, 3, 4, 5, 6 is good, because = 4+6. Find the other solution! Which is the smallest one from the five numbers? (A) 1 (B) 3 (C) 4 (D) 5 (E) If we divide 948 and 417 with the same 2-digit number, the remainder will be the same. What will be the quotient, if we divide the 948 with this 2-digit number? (A) 55 (B) 32 (C) 23 (D) 16 (E) 86 Cadet Which digit is on the first place in the smallest integer, which multiply with 2013 ends on 2012? (A) 7 (B) 4 (C) 1 (D) 3 (E) Integers m and n satisfy (6 m)(6 + n) = 12. How many such m s exist? (A) 6 (B) 7 (C) 12 (D) 13 (E) None of the previous Four rectangles built a rectangle with a rectangular hole inside. Three of the rect- 75

76 Cadet, 4 point problems angles are shown in the pictures. The fourth rectangle you may choose yourself. In the picture you see an example with a hole of two little squares. Ann chooses the fourth rectangle to get the biggest hole that is possible. How many little squares fit in that hole? (A) 4 (B) 6 (C) 10 (D) 12 (E) A path has white, grey and black tiles in a regular order. Ann walks along the path, from the left to the right. At each tile she mentions the number of white, grey and black tiles she had passed already. w is the number of white tiles, g is the number of grey tiles, b is the number of black tiles. What is always true? (A) b g and b w (B) b w g (C) w g b (D) w b g (E) g w b Cadet In the large picture the grey area is bigger than the white area. The difference is some number of squares as in the small picture. How much of these small squares is the difference? (A) 17 (B) 21 (C) 25 (D) 29 (E) The teacher asked his 25 pupils about their hobbies. Each of the pupils has at least one of the hobbies music, art or sport. 3 pupils only loves sport, 4 only loves music, 5 only loves art. 6 pupils loves sport and music, 7 loves sport and art and 8 loves music and art. The teacher forgot to ask who loved all three of those hobbies. How many pupils loves all three of the hobbies? (A) 0 (B) 1 (C) 2 (D) 3 (E) Ann has written the letters K, A, N, G, O, E, R, O, and E (kangoeroe = kangaroo in 76

77 Cadet, 4 point problems Dutch) on nine different tiles. The tiles are placed in the wrong order. Ann want to get them in the right order by pair wise interchanging two tiles which are neighbours (next to each other). What is the minimal number of interchanges she has to make? (A) 25 (B) 26 (C) 27 (D) 28 (E) In a 6x8-grid half of the squares are not intersected by one of the diagonals, as you can check in the picture. How many of the squares in a 6x10-grid are not intersected? (A) 28 (B) 29 (C) 30 (D) 31 (E) Different letters represent different digits, same letters represent the same digits. For example, AABAB can denote the number If AB + BA = 187, what is the result of A B? (A) 0 (B) 16 (C) 17 (D) 56 (E) 72 Cadet We use black and white circles to build triangles such as in the picture. Every odd-numbered layer consists of black and white circles, alternating, starting with a black one. Every even-numbered layer is completely white. How many circles does the smallest such triangle with 10 black circles have? (A) 10 (B) 21 (C) 28 (D) 36 (E) Three brothers want to visit a fair. Unfortunately, their amount of money seems to be insufficient. Albert has 7 euro, Bert has 11 euro and Chris 17 euro. Their father gives the three boys 10 euro together, after which the boys decide to divide all the money evenly. As a result Chris has less money than before. How much? 77

78 Cadet, 4 point problems (A) 1 EUR (B) 2 EUR (C) 3 EUR (D) 4 EUR (E) 5 EUR Michelle has 4 skirts: a blue, a red, a green and a black one. She has 4 blouses: a blue, a red, a green and a black one. Furthermore, she has 4 pullovers, a blue, a red, a green and a black one. In the middle of the night, she wants to take some clothes from the cabin without switching the lights on. She can t feel the difference between a skirt, a blouse and a pullover. How many clothes does she have to take to be certain that she gets a fitting collection of clothes (i.e. of the same colour)? (A) 3 (B) 4 (C) 6 (D) 9 (E) The edges of rectangle ABCD are parallel to the coordinate-axes. ABCD lies below the x-axis and on the right of the y-axis (i.e. in the fourth quadrant), see the figure. The coordinates of the four points A, B, C and D are all integers. We calculate for each of these points the number y-coordinate x-coordinate. Which of the four points gives the smallest number? Cadet (A) A (B) B (C) C (D) D (E) depends on the rectangle John has made a building of cubes. In the picture you see this building from above. In each cell you see the number of cubes in that peculiar tower. When you look from behind, what do you see? (A) (B) (C) (D) (E) 78

79 Cadet, 4 point problems This table with 12 cells is made by drawing 9 lines (5 horizontally and 4 vertically). Using 9 lines in a different way, 5 horizontally and 4 vertically, you will get a table with only 10 cells. What is the maximum number of cells you can get if you draw 15 lines? (A) 22 (B) 30 (C) 36 (D) 40 (E) An hairdresser cuts the hair of only boys and men. It takes him exactly 12 minutes cutting the hair of a boy and exactly 20 minutes for the hair of a man. Today he has worked for exactly 8 hours. He has cut the hair of at least 7 boys. The hair of how many men was cut by the hairdresser at most today? (A) 10 (B) 17 (C) 18 (D) 24 (E) In the diagram, triangle P QR is equilateral, SR bisects P RQ, and T is a point on RS such that T QR : T QS = 2 : 1. What is the size of ST Q? (A) 40 (B) 50 (C) 60 (D) 70 (E) The clown was thinking of a colour. Her friends tried to guess the colour. Their guesses where - You are thinking either red or green. - You are thinking either green or blue. - You are thinking either red or yellow. - You are thinking either green or yellow. If only one guess was correct, what colour was the kangaroo thinking of? (A) red (B) green (C) blue (D) yellow (E) the situation described is impossible Cadet If 2 x = 3, what is the value of 16 x 3? (A) 6 (B) 13 (C) 24 (D) 78 (E) none of the previous On the surface of the globe there have been drawn 10 parallels and 10 meridians. How many parts has the surface been broken into with these lines? (A) 81 (B) 90 (C) 100 (D) 110 (E) At each break between lessons Maxim eats a candy. In five days there have been 32 lessons. How many sweets has Maxim eaten within these days? (A) 24 (B) 25 (C) 27 (D) 31 (E) 32 79

80 Cadet, 4 point problems In the triangle ABC, the median BE is perpendicular to the bisector AD. Find the length of AB if AC = 12cm. (A) 4 cm (B) 6 cm (C) 7 cm (D) 8 cm (E) 12 cm The checked square 4 4 has all 25 mesh points marked. How many different lines, each of which passes through at least three marked points are there? (A) 10 (B) 12 (C) 20 (D) 32 (E) Andriy and Michael had two identical rectangles. Andriy cut his rectangle into figures like L (see fig.) while Michael cut his into rectangles measuring 2 3. What is the least length of the larger side of the initial rectangle? The length of a small square is 1. (A) 5 (B) 6 (C) 8 (D) 12 (E) 15 Cadet The logo of the company GEO is a coloured circle with the radius 4 cm with a white square inside (as in the picture). How large is the coloured area approximately? (A) 18 cm 2 (B) 16 cm 2 (C) 14 cm 2 (D) 9 cm 2 (E) 5 cm The cyclist covered 25 km distance at approximate speed of 20 km/h. How long did the journey take? (A) 1 h 25 min. (B) 1 h 15 min. (C) 1 h 5 min. (D) 1 h (E) 48 min In two months part time job Alex earned a sum rounded off to 1100 EUR. If he earned only 3 EUR less, the sum would be rounded off in the same way to 1000 EUR. What is the maximum sum Alex could have earned? (A) 1047 (B) 1049 (C) 1050 (D) 1052 (E) If I take my age, subtract one, multiply the result with my daughters age I get the answer In which year was my daughter born? (A) 1950 (B) 1951 (C) 1981 (D) 1980 (E) more information is needed Roo likes long jumps. He never jumps less than 5 meters. In front of him there are three bushes that Roo would like to eat shoots from. Two of them area at a distance of 5 meters 80

81 Cadet, 4 point problems from Roo and there is a distance of 5 meters between them. The third one grows exactly in between the two. What is the minimal number of jumps Roo has to take in order to reach all the three bushes and return to the place he is now? (A) 3 (B) 4 (C) 5 (D) 6 (E) In a school there are 30 classes. All classes have 5 sport lession a week. The school has only 2 gyms. The lessions are from monday until friday, every the the same number of lessions in both gyms. How hany lessions need to organize in a gym pro day, if all sportlessions want to organize in the gyms? (A) 5 (B) 10 (C) 15 (D) 20 (E) A die is a cube with its faces numbered from 1 to 6 (with dots). Numbers on opposite faces add up to 7. A die lays on a plane as shown in the figure. A move consists of a rotation of 90 degrees around one of the edges of the lower face, until a new face touches the plane. There are four possible moves named N, W, S and E, according as the die moves to the North, West, South or East, respectively. If the sequence of moves W, N, E, S is repeated 2013 times, how many points are there on the upper face in the final position of the die? (A) 1 (B) 2 (C) 3 (D) 5 (E) other The boundary of the region in the picture is the union of half-circles, each of radius 10 cm, such that any two adjacent of them have orthogonal diameters. What is, in cm2, the area Cadet of the region? (A) 1600 (B) 400π (C) 350π (D) 300π (E) A different answer In the squares of the 5 5 chessboard integers are written so that the numbers in adjacent squares differ by 1. We also know that numbers 3 and 11 are written there. How many different numbers are denoted in the table? (A) 7 (B) 8 (C) 9 (D) 10 (E) After reconstruction of the equipment labour productivity of workers at a factory has grown on 25 %. Therefore the administration has decided to dismiss 20 % of workers. On how 81

82 Cadet, 4 point problems many percent the volume of production has changed after that? (A) has reduced on 5% (B) has reduced on 2,5% (C) has reduced on 2% (D) not changed (E) has increased on 5% Different letters correspond to different digits and similar letters correspond to similar digits: K A N G A R O O = Find A + O. (A) 5 (B) 6 (C) 7 (D) 9 (E) A two-minute sandgalss has been built of two similar cubes. When all sand is in one cube then an half of the cube has been filled. If the sand in the sandglass falls 20 seconds without a breake the level of the sand in the lower cube ascends exactly 1 cm. Find volume of the sand in the sandglass. (A) 4 3 cm 3 (B) 6 3 cm 3 (C) 83 2 cm3 (D) 10 3 cm 3 (E) cm A series of positive integers has the property that each member of the series, except the first one, is equal to the sum of the digits of the previous member. If the first member is and the last is one-digit number, find the last member. (A) 0 (B) 1 (C) 2 (D) 4 (E) An electronic calendar shows the dates in the format dd.mm.yyyy as in the example How many times in the period the digits of the day and the month are exactly the digits of the year? (A) 51 (B) 52 (C) 53 (D) 55 (E) 57 Cadet Ann, Bob and Clara participated in Kangaroo contest. Bob received 50% points less than Ann. Clara received 50 % points more, than Bob. Ann, Bob and Clara received 270 points in total. How many points received Clara? (A) 90 (B) 100 (C) 110 (D) 120 (E) Height BH and bisector AL of triangle ABC are intersected at a point M. HML = 100, MLC = 110. Find the greatest angler of triangle ABC. (A) 80 (B) 90 (C) 100 (D) 110 (E) Peter wrote down 10 integers. Then he calculated all possible pairwise products of these numbers. It appeared that exactly 20 products are negative. How many products can be positive? (A) 10 (B) 16 (C) 20 (D) 25 (E) impossible to determine For what greatest n one can receive n-gon as a figure of intersection of a convex quadrangle and a triangle? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 82

83 Cadet, 4 point problems Winnie Pukh wants to leave the house (marked V on the scheme), to visit all his friends, to pass each path connecting their houses exactly once and to return back to his home. On what path he needs to leave the house? (A) 1 (B) 2 (C) 3 (D) 4 (E) it is impossible to do Andy, Betty, Cathie, Dannie and Eddy were born on 20/02/2001, 12/03/2000, 20/03/2001, 12/04/2000 and 23/04/2001 (day/month/year). Andy and Eddy were born in the same month, Betty and Cathie were born in the same month as well. Andy and Cathie were born in the same day of different months, Danny and Eddy were born in the same day of different months as well. Who of these children is the youngest? (A) Andy (B) Betty (C) Cathie (D) Dannie (E) Eddy What of following figures can be bent so, that the surface of a cube will turn out exactly? (A) all (B) only 1, 2, 4 (C) only 1, 2, 3 (D) only 1, 4, 5 (E) only 1, 2, 4, There are 7 identical cubes and 7 identical balls. If the weight of 7 cubes together with 6 balls is 71 g but the weight of 6 cubes together with 7 balls is 72 g how much is the weight of one cube? (A) 5 g (B) 6 g (C) 7 g (D) 8 g (E) 9 g Cadet Julian has written 4 numbers, the second he wrote is 1 less than the double of the first number, the third is 2 less than the triple of the second number and the fourth is 3 less than the fourfold of the third number. The sum of the 4 numbers is 631. What is the first number? (A) 16 (B) 17 (C) 19 (D) 20 (E) Marie has formed a cube from plasticine. With a knife she has made a plane cut through the cube. The cutting surface that arises this way cannot be (A) an equilateral triangle (B) a parallelogramm (C) a hexagon (D) a rectangular triangle (E) a pentagon Violet and her two brothers collected wood for a bonfire. Her younger brother found half as much as Violet and her older brother found three times as much as Violet. All together 83

84 Cadet, 4 point problems they collected 22.5 kg of wood. How much did Violet collect? (A) 1 kg (B) 3 kg (C) 4 kg (D) 5 kg (E) 7 kg Carina and Alex play a game of battleship on a 5 5 board. Each one of them puts the three ships, and on their board such that no two ships have a common point. Carina already put 2 ships down. How many positions are there for her 3 1 ship? (A) 4 (B) 5 (C) 6 (D) 7 (E) In a circle of radius 3 cm we inscribe a rectangle ABCD and call I, J, K, L the midpoints of its sides. What is, in cm, the perimeter of the rhombus IJKL? (A) 6 (B) 9 (C) 12 (D) 4 3 (E) It depends on the shape of rectangle My rectangular field has the same area as a square of side 70 m. If I reduced its length by 1 m and increased its width by 2 m, it would keep the same area. What is the length of my field? (A) 49 (B) 68 (C) 72 (D) 98 (E) 100 Cadet How many times does the digit 0 occur when 5 5 is written out in full? (A) 0 (B) 1 (C) 2 (D) 3 (E) Four of the following numbers are prime; the other is divisible by 17. Which is the multiple of 17? (A) 331 (B) 3331 (C) (D) (E) On the map of England in my atlas, the point representing Oxford is 6cm from the point representing Cambridge. The scale of the map is given as 1: To the nearest 10 kilometres what is the distance from Oxford to Cambridge? (A) 20 km (B) 90 km (C) 100 km (D) 110 km (E) 1010 km Given that m is even and n is odd, what is the units digit of 5m + 5n? (A) sometimes 5 and sometimes 0 (B) neither 5 nor 0 (C) 0 (D) 5 (E) more information needed A rectangle has been divided into two squares, each of perimeter 36 cm. What was the perimeter of the rectangle? (A) 36 cm (B) 48 cm (C) 54 cm (D) 60 cm (E) 72 cm 84

85 Cadet, 4 point problems A square number is divided by 10. Which of the following could be the remainder? (A) 2 (B) 3 (C) 6 (D) 7 (E) How many three-digit positive integers are multiples of both 5 and 9? (A) 10 (B) 18 (C) 20 (D) 25 (E) The sum of natural numbers a, b, c, and d is In the sequence a, b, c, d, each of the numbers b, c, d either is twice its antecedent, or is equal to it. Find the number a. (A) 61 (B) 157 (C) 183 (D) 503 (E) such a, b, c, d cannot exist When discussing his new government (consisting of at least 5 persons), the primeminister said that among any 5 ministers there are exactly 4 new persons. How many ministers can be there in his government? (A) only 5 (B) only 6 (C) any number exceeding 9 (D) any number exceeding 4 (E) such a government cannot exist Vasya wrote down several consecutive natural numbers. The percent of odd numbers among them cannot be (A) 40 (B) 45 (C) 48 (D) 50 (E) The middle of the second third of an hour does not coincide with (A) the end of the first half of this hour (B) the middle of this hour (C) the end of the second quarter of this hour (D) the middle of the third quarter of this hour (E) this moment coincides with all the time moments (A) - (D) We consider all the angles depicted in the figure (each of them are less than 180 ) and draw all the bisectors of these angles. How many rays shall we see at the new figure? Cadet (A) 11 (B) 12 (C) 13 (D) 15 (E) The perimeter of a trapezoid is 5 and the lengths of its sides are integers. What are its base angles? (A) 30 and 30 (B) 60 and 60 (C) 45 and 45 (D) 30 and 60 (E) impossible to determine If the shortest side of an isosceles triangle is equal to its longest height, then this triangle (A) is necessarily acute-angled 85

86 Cadet, 4 point problems (B) is necessarily a right triangle (C) is necessarily obtuse-angled (D) can be acute-angled and can be obtuse-angled (E) can be acute-angled and can be right-angled We say that a triangle ABC is normal if a new triangle can be formed by the three heights of ABC. How many triangles in the figure are not normal? (A) 1 (B) 2 (C) 3 (D) 4 (E) Kate has 2 euros. She decided to earn more money. She invested that 2 euros, and after one turn she had 4 euros. Then she saved 2 euros, and invested 2 euros again. After every turn her investment is doubled. After every second turn (not counting the first), she saved the invested sum (from the last turn). How many euros will she have in total after the 10th turn? (A) 20 (B) 40 (C) 64 (D) 126 (E) All the friends of Ashley the dog are either Labrador retrievers or golden retrievers. 2/7 of her friends are Labrador retrievers, 1/3 of her friends are males and 1/6 of her friends are male Labrador retrievers. Which portion of Ashley s friends are female golden retrievers? (A) 5/7 (B) 2/3 (C) 5/6 (D) 23/42 (E) none Cadet Define an operation by a b = a + b + ab (for example 2 3 = = 11). The result of ( s) is (A) (B) (C) (D) (E) other Ann and Ben play the following game: Ann says a positive integer a, then Ben says the greatest divisor b of a different from a, then Ann says the greatest divisor c of b different from b, and so on. The first who says a prime number wins. Which of the following numbers should choose Ann to start the game and win? (A) 60 (B) 64 (C) 72 (D) 84 (E) none of them From a list of three numbers we call changesum the procedure to made a new list replacing each number by the sum of the other two. For example, from {3, 4, 6} changesum gives {10, 9, 7} and a new changesum leads to {16, 17, 19}. If we begin with the list {20, 1, 3}, what is the result of subtracting the first number of the list minus the second after 2013 consecutive changesums? (A) 17 (B) 19 (C) 17 (D) 19 (E) another result Admission fees in the auditorium of Science Park are 75 euro cents for adults and 25 euro cents for young people. One afternoon in the auditorium of the Park, with a capacity for 600 people and it was not full, was collected 300 euros. How many adults attended at least? (A) 359 (B) 300 (C) 365 (D) 361 (E)

87 Cadet, 5 point problems We fill a grid of p rows and q columns with all the integers from 1 to p q, each number once. We write it in increasing order, filling first row 1, after row 2, etc.. If the number 20 is in the third row, number 41 in the 5th row and number 103 in the last row, what is the value of p + q? (A) 21 (B) 22 (C) 23 (D) 24 (E) point problems On a wooden cube little Tom drew some line segments connecting opposite vertices of its sides (at most one segment on each side). Let n be the number of the cube s vertices at which two or three of the drawn segments meet. How many different possible values of n are there? (A) 8 (B) 7 (C) 9 (D) 5 (E) In the expressions K + A + N + G + A + R + O + O, C + O + O + L and A + O the letters denote non-zero digits (different letters denote different digits). All three sums divide the number How many different digits can the letter A denote? (A) 1 (B) 2 (C) 3 (D) 4 (E) There are no such digits A die is a cube with a different number of points on each face, from 1 to 6. A die is rolled over a table and the points on all its visible faces are added and written on a piece of paper. This action is repeated one or more times. Finally, all the written numbers are added to obtain a result. Which is the largest integer that cannot be obtained as a result? (A) 29 (B) 35 (C) 44 (D) 79 (E) other How many positive integers not greater than 100 may be written in the form a b, where a, b are integers, both greater than 1? (A) 10 (B) 12 (C) 14 (D) 15 (E) Assume that x 2 y 2 = 84, where x and y are positive integers. How many values may the expression x 2 + y 2 have? (A) 1 (B) 2 (C) 3 (D) 5 (E) 6 Cadet Grandma s vegetable garden has been divided into four zones. The surface area values of three of them are given in the picture. What is the area of the fourth zone? (A) 6 (B) 7 (C) 20/7 (D) 28/5 (E) 35/ A two-digit number is added to its reverse (a two-digit number too) and to the sum of its digits. What is the maximum number of different results which may be obtained starting 87

88 Cadet, 5 point problems with various two-digit numbers? (A) 0 (B) 4 (C) 12 (D) 15 (E) If 3 x = a and a y = 81, what is the value of the product xy? (A) 4 (B) 3 (C) 12 (D) 0 (E) If the three-digit number abc is divided by 8 and S = 4a + 2b + c, then the following statement is true for any natural number k (A) S = 8k + 1 (B) S = 8k + 5 (C) S = 4k + 3 (D) S = 4k (E) S = 6k What figure does letter D replace, knowing thar different letters correspond to different figures and identical letters correspond to identical figures? BDCE + BDAE = AECBE (A) 2 (B) 3 (C) 6 (D) 7 (E) The grandmother won an amount of money at the lottery and wants to give it as a present to her three grandchildren. She plans to split it according to the following rule: half of the amount will be split in equal parts, the remaining half will be split proportionally to the age (in an integer number of years) of thegrandchildren. Doing so, the oldest of them, whose age agrees with the sum of the ages of the others, should receive 40 euros. How many euros did the grandmother win at the lottery? (A) 60 (B) 68 (C) 80 (D) 110 (E) A different number Cadet There is only one street connecting Mark s house to Liza s. Every day, driving at constant speed, Mark leaves his house, get Liza s house at 9.00 a.m. and drives Liza back to his house. Last night time has been changed from the legal (summer) to the sun (winter) one, but Liza didn t realize that, so this morning she didn t see Mark at the time she believed the usual one. Then she started to walk in the direction of Mark s house. After some time she met Mark s car coming: immediately the car stopped, she entered the car and Mark drove back to his house, where they arrived 12 minutes earlier than usual. Consider that no time was lost in these operations. How many minutes did Liza walk? (A) 6 (B) 12 (C) 24 (D) 48 (E) Statistics say that only 1 over 14 young people remains without job within 6 months after taking his master degree in Mathematics. Let p the percentage of young people who get some job within 6 months after taking their master degree in Mathematics. What is the 2013th digit of the decimal part of p? (A) 7 (B) 1 (C) 4 (D) 2 (E) A clock has only the hour hand as the minute hand has fallen off. If the hour hand points towards the number 28 of the dial, what is the exact time? (should include a drawing of a clock with one hand, and pointing at 28) (A) 2:04 (B) 5:28 (C) 5:36 (D) 5:58 (E) none of the previous 88

89 Cadet, 5 point problems For three kangaroos named Kan, Ga and Roo it is known that -either Kan or Ga is the tallest of the three, -either Roo is the tallest or Kan is the shortest. Who is the tallest and who is the shortest? (A) Ga is the tallest and Kan is the shortest (B) Roo is the tallest and Kan is the shortest (C) Ga is the tallest and Roo is the shortest (D) Kan is the tallest and Ga is the shortest (E) it is impossible to be sure Mom bakes six pies: first an apricot pie (1), then a strawberry pie (2), then a cherry pie (3), a mushroom pie (4), a pie with jam (5) and a raspberry pie (6). While she does this, children sometimes run into the kitchen, and each time eat the hottest cake. In what order cannot the pies be eaten? (A) (B) (C) (D) (E) There are five numbers given: a 1 = 1, a 2 = 1, a 3 = 1, a 4 = 1, a 5 = 1. The sixth number equals the product of the first to the second, seventh - the product of the second to the third, the eighth - the product of the third to the fourth, etc. What is the sum of the first 2013 of these numbers? (A) 1006 (B) 671 (C) 0 (D) 671 (E) How many unequal triangles are there whose lengths are positive integers and the perimeter is 20? (A) 6 (B) 7 (C) 8 (D) 9 (E) Nick, Ryan, Simon and James from the group Dandelion had their first concert and sang a few songs. There were always three of them singing and the fourth one playing the guitar. James sang 8 songs, which was the most, Ryan the least songs - only 5. How many songs did Dandelion sing at their first concert? (A) 7 (B) 8 (C) 9 (D) 13 (E) 26 Cadet The picture shows how a rabbit was running when a wolf was chasing it in the meadow. At first, the rabbit was heading exactly to the east, then turned right and after a while turned left. Afterwards he turned left one more 89

90 Cadet, 5 point problems time and ran exactly to the east again. What is the size of the angle at the second turn? (A) 48 (B) 82 (C) 88 (D) 90 (E) Little Inuits were competing in building the tallest igloo. Before the contest the old Inuit Amak said that if no participants build igloos with the same height, the competition can have 24 different results. How many Inuits were competing? (A) 2 (B) 3 (C) 4 (D) 5 (E) If a = 2b, c = a + 45 and a + 3b + 2c = 675 then b = (A) 60 (B) 65 (C) 75 (D) 130 (E) There are 6 points on the plane, there is not 3 collinear of them. We connect points with segments. The segments do not cross eachother. Maximum how many segments is possible to draw this way? (A) 9 (B) 10 (C) 11 (D) 12 (E) There is a rectangle on the plane, divided into 1 1 squares. How many squares are crossed by a diagonal of the rectangle? (A) 4022 (B) 4023 (C) 4024 (D) 4025 (E) A clock makes 3 strokes per 4 seconds. In how many seconds will it make nine strokes? (A) 10 (B) 12 (C) 14 (D) 16 (E) 18 Cadet From point O there has been drawn a ray, at an angle of 10 clockwise there has been a second ray drawn to it, then the third ray has been drawn at an angle of 20 clockwise, etc. (each next corner is twice as large as the previous one). How many different rays can be drawn this way? (A) 6 (B) 8 (C) 10 (D) 16 (E) infinitely many In the squares of the 5 5 square board integers are written so that the numbers in adjacent squares differ by 1. We also know that numbers 3 and 11 are written there. Find the sum of all the numbers placed in the table. (A) 175 (B) 140 (C) 105 (D) 90 (E) parrots are talking: First: The second parrot is green ; Second: The third parrot is green ; th parrot: 2012th parrot is green ; 2012 th parrot: 2013th parrot is a blue hippopotamus. 2013th parrot: I am not a blue hippo! It is known that all green parrots lie and all parrots that lie are green. How many parrots are green? (A) 1 (B) 2 (C) 1006 (D) 1007 (E) Find the largest possible value of the product of several natural numbers if the sum of these numbers is (A) (B) (C) (D) (E) Andy, Bobby, Charley and Davy are schoolmates. Andy is 60 days older then Bobby, Charley is 80 days older then Davy, and Bobby is 100 days younger then Charley. Andy s 90

91 Cadet, 5 point problems birthday is March, 10th. Determine Davy s birthday (A) March, 30th (B) April, 19th (C) May, 19th (D) January, 9th (E) impossible to determine Three painters should paint 20 identical cubes. Each side of a cube should be painted completely only by one painter. Different painters can paint different sides cube, but no two of them can paint the same cube simultaneously. For painting of any side of a cube 10 seconds are required to each painter. Painters begin to work simultaneously. For what least time painters can paint completely all 20 cubes? (A) 380 seconds (B) 390 seconds (C) 400 seconds (D) 410 seconds (E) 420 seconds Marathon races took place in some school. It has appeared on the finish, that Andrew outstripped 2 times more participants, than outstripped Daniel, and Daniel outstripped 1,5 times more participants, than outstripped Andrew. Andrew occupied 21th place (i.e. he was 21th on the finish). How many runners have participated in a marathon? (A) 31 (B) 41 (C) 51 (D) 61 (E) In a class with 30 students only 12 have mobile phone. During Christmas holidays the students that do not have mobile phone send postcards to all colleagues, and the students with mobile send sms to the colleagues with mobile phone and postcards to the colleagues that do not have mobile phone. How many postcards are sent? (A) 132 (B) 216 (C) 522 (D) 728 (E) In 2013 the sum of the digits is 6. How many times this happened between 1000 and 2000? (A) 4 (B) 9 (C) 15 (D) 18 (E) John chooses a 5-digit positive integer and deletes one of its digits. Find the sum of the digits of the chosen integer if the sum of the 5-digit and the 4-digit numbers is equal to (A) 26 (B) 20 (C) 23 (D) 19 (E) 17 Cadet The side of the unit squares of the net is equal to 2 cm. Find the area in sq. cm of the quadrilateral ABCD. (A) 96 (B) 84 (C) 76 (D) 88 (E) Using the digits 1, 3, 5 and 7 Charley made three various four-digit numbers (in each number each of these digits used exactly once). To what of the following values the sum of these three numbers cannot be equal? (A) (B) 4665 (C) (D) 6861 (E) Each of four verticies and six edges of a tetrahedron is marked with ten numbers 1, 2, 3, 91

92 Cadet, 5 point problems 4, 5, 6, 7, 8, 9 and 11 (number 10 is omitted), any of the ten number listed being used exactly once. It is known that for any two verticies of the tetrahedron the sum of two numbers which mark these verticies equals the number which mark the edge connecting these two verticies. The edge AB is marked with the number 9. With which the number the edge CD is marked? (A) 4 (B) 5 (C) 6 (D) 8 (E) What the maximal quantity of numbers it is possible to arrange in a row in order to the sum of any of three successively going numbers be negative, and the sum of any five successively going numbers be positive? (A) 5 (B) 6 (C) 8 (D) 10 (E) it is impossible to do In the given figure CAD = 11, AB = OC, where O is a center of the circle. Find COD. (A) 30 (B) 33 (C) 35 (D) 40 (E) 45 Cadet trees (maples and lindens) grow along avenue in park. The number of trees between any two maples does not equal 3. What the greatest number of maples can be among these 20 trees? (A) 8 (B) 10 (C) 12 (D) 14 (E) the situation is not possible In each cell of a 4 4 board one kangaroo sits on. At midday every kangaroo jumps over the side of a cell on the next cell. What greatest number of cells can appear empty? (A) 6 (B) 8 (C) 9 (D) 10 (E) Peter wrote numbers from 1 up to 9 into circles in the following figure so that the sum of three numbers along each line is divisible by 5. What number can be in the center? 92

93 Cadet, 5 point problems (A) only 2 (B) only 5 (C) only 8 (D) 2, 5 or 8 (E) any from 1 up to Some schoolboys visit a study on solving of the Kangaroo problems. Each of 4 of these schoolboys has no more 3 friends among participants of study. Each other participant has at least 7 friends among participants of study. What least number of schoolboys takes part in Kangaroo study? (A) 8 (B) 10 (C) 12 (D) 16 (E) Five people attend the first lesson in driving school. Among them there is one person that is friends with exactly 3 of the others and there are 3 people that are friends with exactly 2 of the others. How many friends does the fifth person have among the others? (A) 1 or 2 (B) 1 or 3 (C) 1, 2 or 3 (D) 2, 3 or 4 (E) 2 or Clara has a game with 6 buttons that glow either red or blue. If she presses a button then this button and all buttons that touch this button with a side change their colour. At the moment the buttons look as shown in the picture. If Clara presses the buttons C, D, E and F all buttons will glow red. However, she could also press less than four buttons to get the same result. Which ones? (A) B, C and D (B) A, E and F (C) A and E (D) only F (E) A and F When Rick and Rose found their old model railway, Rick quickly made a perfect circle from 12 identical track parts. With the same type of track parts Rose tries a little more difficult track starting with 2 of these pieces. She wants to use as few pieces as possible for her closed track. How many pieces does her track consist of? Cadet (A) 16 (B) 18 (C) 21 (D) 22 (E) The roundabout shown in the picture is entered by 4 cars at the same time, each one from a different direction. Each of the cars drives less than one round and no two cars leave the roundabout in the same direction. How many different combinations are there for the cars leaving the roundabout? 93

94 Cadet, 5 point problems (A) 9 (B) 12 (C) 15 (D) 24 (E) What is the value of ? (A) 2 5 (B) 5 12 (C) 12 5 (D) (E) Among the figures (A) - (D), which one cannot be used as a tile to pave the plane (without overlapping)? (A) (B) (C) (D) (E) the plane can be paved by any one of the figures (A) - (D) Vasya has drawn a square ABCD with side 2013 and marked a point Eon the side AB so that AE = 1. Petya has drawn a rectangle KLMN such that KL = 1 and LM = n. The boys discovered that ACE = KML. What was the value of the number n? (A) 2013 (B) 2014 (C) 4024 (D) 4025 (E) Eight teams play a volleyball tournament. Every two teams meet, and there is no draw games in volleyball. Looking at the tournament results, Victor found 5 teams with exactly k victories each. What is the maximal possible value of k? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 Cadet An angle bisector cuts the opposite side of a triangle in the ratio 1 : 2. What is true then? (A) This triangle must be isosceles. (B) This bisector must pass through the middle of one of the medians. (C) This triangle must be right (D) In this triangle one angle must be twice some other angle. (E) Any statement (A) - (D) may fail Peter has measured 4 sides and 2 diagonals of a quadrangle ABCD. Among the 6 numbers obtained, there are 3 pairs of identical numbers but there is no triple of identical numbers. Then necessarily (A) ABCD is a rectangle (C) ABCD is an isosceles trapezoid (E) any statement (A) - (D) may fail (B) ABCD is a rhombus (D) the diagonals of ABCD are equal A natural number N is smaller than the sum of its 3 greatest divisors (naturally, excluding n itself). What statement must be true? (A) N is divisible by 4 (B) N is divisible by 5 (C) N is divisible by 6 94

95 Cadet, 5 point problems (D) N is divisible by 7 (E) there is no such N We denote by A the number of squares among the integers from 1 to , and by B the number of cubes among the same integers. Then (A) A = B (B) 2A = 3B (C) 3A = 2B (D) A = 2013B (E) A 3 = B Basil is encoding numbers as follows: starting with an integer, he writes down the product of its first two digits, then he writes the product of the second and the third digits, and so on. For instance, the number 5648 will turn to Which of the following numbers cannot be the resulting code? (A) 634 (B) 2012 (C) 3283 (D) 6454 (E) There are two clocks on the wall. One of them shows the precise time, while the other clock runs fast. Now the angle between the hour hands of these clocks is 42. What is the angle between their minute hands? (A) 144 (B) 120 (C) 84 (D) 21 (E) The points D, E, and F are marked on the sides of the triangle ABC as shown in the figure. The triangles ACD, ADE, AEF, and EF B are equal to each other. What is the value of the angle ABC? (A) 10 (B) 15 (C) 30 (D) 45 (E) We say that a nonnegative integer n is rich if the sum of all its divisors (except n itself) is greater than n. For example, the number 12 is rich because > 12. Integers of what kind cannot be rich? (A) perfect squares (B) multiples of 2013 (C) a number exceeding (D) powers of 3 (E) integers of all kinds (A) - (D) can happen to be rich Cadet 95

96 Junior, 3 point problems 5 Junior point problems The number is not divisible by (A) 2. (B) 3. (C) 5. (D) 7. (E) The eight semicircles build inside the square are congruent and the side of this square has length 4. What is the area of the non-shadowed part of the square? (A) 2π (B) 8 (C) 6 + π (D) 3π 2 (E) 3π Consider a three-quarter circle with center M and an orientation arrow as indicated in the picture on the right. What is the position of the oriented three-quarter circle when it is first rotated counterclockwise by 90 around M and then reflected at the x-axis? (A) (B) (C) (D) (E) In triangle ABC, the measure of angle BAC is 120, AC = 2AB and M is the midpoint of BC. What is the true relation? (A) m(mac) = 90 (B) m(mab) = 60 (C) m(amc) = 120 (D) m(amb) = 30 (E) m(mab) = 90 Junior Which pair is a solution of the inequality 3x xy 3y 2? (A) (2, 0) (B) (1, 1) (C) (9, 2) (D) (0, 2) (E) ( 1, 1) Adding 4 15 to 8 10, Mihai has also obtained a number which is a power of 2. Find this number (A) 2 10 (B) 2 15 (C) 2 20 (D) 2 30 (E) If O is the centre of the circle and OB = BC, what relation between x and y is always true? 96

97 Junior, 3 point problems (A) 2x = 3y (B) x + y = 90 (C) x = 2y (D) x + 2y = 180 (E) x = y Consider the equilateral triangle with side equal to 1, and the squares based on its sides. Find the length of segment AB (A) 1 + 3/2 (B) (C) (D) (E) Kant the Ant leaves A to get to B, following the arrow indicators. Find the path she needs to cover in order to obtain, in the end, the minimum result of the operations she meets. (A) 1 (B) 2 (C) 6 (D) 1 (E) On the outside a cube is painted with black and white squares as if it was build of four white and four black smaller cubes. Which of the following is a correct building scheme for this cube? Junior (A) (B) (C) (D) (E) 97

98 Junior, 3 point problems The sum of a number and its reciprocal is Which of the following numbers can this number be? (A) 0.6 (B) 0.75 (C) 0.8 (D) 1.0 (E) Bert runs the 100 meters in 10,2 seconds. The same distance takes his running friend Ernie 12 seconds. A start of how many meters does Ernie have to get so that the two friends can start and finish at the same time? (A) 10 (B) 12 (C) 15 (D) 18 (E) is divisible by 11 and is made of the four consecutive digits 0, 1, 2 and 3. What is the smallest next number which is divisible by 11 and made of four consecutive digits? (A) 2031 (B) 2103 (C) 2130 (D) 2134 (E) Let a be a real number that satisfies a 3 = a + 1. Consider the following statements: a 4 = a 2 + a, a 4 = a 5 1, a 4 = a 3 + a 2 1, a 4 = 1. How many of them are correct? a 1 (A) 0 (B) 1 (C) 2 (D) 3 (E) Three of the numbers 2, 4, 16, 25, 50, 125 have product What is their sum? (A) 70 (B) 77 (C) 131 (D) 143 (E) None of the previous Mrs. Margareth bought 4 cobs of corn for everyone in her 4-member family. In the shop she got the discount the shop offered. How much did she pay? (A) 0,80 (B) 1,20 (C) 2,80 (D) 3,20 (E) The Scotts family apartment plan is made in scale 1 : 50 and has a rectangular shape with dimensions 20 cm 30 cm. What is the area of the apartment? (A) 12 m 2 (B) 150 m 2 (C) 300 m 2 (D) 450 m 2 (E) 600 m 2 Junior George started training for the marathon 18 weeks before the run. According to the plan, he had to run 60 km each week. Unfortunately, during the 12th week he got sick and was not training at all. How many kilometers does he need to run during the remaining 6 weeks to catch up with the original plan? (A) 10 (B) 70 (C) 78 (D) 360 (E) How many terms need to be included in the calculation for the result to be equal 2013? (A) 2011 (B) 2012 (C) 2013 (D) 4026 (E) The result can never equal How many decimal places are there in the decimal number ? (A) 10 (B) 12 (C) 13 (D) 14 (E)

99 Junior, 3 point problems This year John can multiplied his age with the age of his son and get the answer In which year was John born? (A) 1981 (B) 1982 (C) 1953 (D) 1952 (E) more information is needed If = 2 n then n = (A) 2027 (B) 1999 (C) 1998 (D) 1980 (E) Let C 1 and C 2 be two different circles with centre O. Let [AB] be a chord of the C 1 that is tangent to C 2 and such that AB = 20cm. What is the area of the grey region? (A) 2πcm 2 (B) 10πcm 2 (C) 20πcm 2 (D) 100πcm 2 (E) 120πcm Which is the smallest natural number n such that the remainder of the quotient of n by either 5, 6 or 9 is 1? (A) 21 (B) 91 (C) 121 (D) 271 (E) Let f be a real-valued function with graph. The graph of the function f(2 x) 1 is (A) (B) Junior (C) (D) 99

100 Junior, 3 point problems (E) How many six-digit numbers are there which contain the string 2013? (A) 278 (B) 280 (C) 282 (D) 290 (E) In Pippa s Pizza Parlour, the cost of a pizza is proportional to its volume. A 15 cm diameter pizza costs 2.99 EUR. What is the cost of a 30 cm diameter pizza with the same thickness? (A) 5.98 EUR (B) 5.99 EUR (C) 8.97 EUR (D) EUR (E) EUR In a record-breaking journey, a female humpback whale travelled across a quarter of the globe, a distance of at least km, according to a news item. On that basis, roughly what is the radius of the earth? (A) 1500 km (B) 3000 km (C) 6500 km (D) km (E) km The number n is the largest positive integer for which 4n is a 3-digit number, and m is the smallest positive integer for which 4m is a 3-digit number. What is the value of 4n 4m? (A) 900 (B) 899 (C) 896 (D) 225 (E) What is the minimal number of chords on a circle such that the number of intersecting points among them in the interior of the circle is exactly 50? (A) 9 (B) 10 (C) 11 (D) 12 (E) In the picture the areas of the triangles are 5 and 10, as shown, and the lines a, b and c are parallel. If the distance between a and c is 6, what is the length of P Q? Junior (A) 4 (B) 5 (C) 6 (D) 36 5 (E) Which of the numbers (A) - (E) is the largest? (A) (B) (C) (D) (E) The numbers 1, 2 and 3 are all roots of the equation x 3 + 4x 2 + x 6 = 0. How many real roots does the equation x 6 + 4x 4 + x 2 6 = 0 have? (A) 1 (B) 2 (C) 3 (D) 4 (E) Six points are marked on a square grid with cell of size 1. What is the smallest area of 100

101 Junior, 3 point problems a triangle with vertices at marked points? (A) 1/4 (B) 1/3 (C) 1/2 (D) 1 (E) Given a six-digit number. The sum of its digits is an even number, the product of its digits is an odd number. Which is the correct statement about the given number? (A) Either two or four digits of the number are even. (B) The number must be odd. (C) The amount of the odd digits of the number is odd. (D) The number can consist of six digits different from each other. (E) Such a number cannot exist An ant wants to walk from point K to point L the shortest way possible. What is the distance the ant has to walk? (A) a + b + c (B) a 2 + b 2 + c 2 (C) a 2 + b 2 + c 2 + 2ac (D) a 2 + b 2 + c 2 + 2bc (E) It depends on a, b and c After simplificating the fraction x3 x 2 y, (y 0, x y), how many of the following y 3 xy 2 expression can be the result: x2 ; x2 ; ( x y 2 y 2 y )2 ; ( x y )2 ; ( x y )2? (A) 0 (B) 1 (C) 2 (D) 3 (E) Which is the greater sum of the divisors of 2013? (A) 75 (B) 95 (C) 194 (D) 2014 (E) 195 Junior Cheli draws the figures shown in the 101

102 Junior, 4 point problems graphic aside. How many of these figures have the same quantity of triangles? (A) 0 (B) 1 (C) 2 (D) 3 (E) The regular octagon of the figure measures 10 cm on each side. Which is the measure of the radius of the circle inscribed in the smallest octagon formed by the diagonals? (A) 10 (B) 7,5 (C) 5 (D) 2,5 (E) We say that an integer number is ascending if each of its figures, from the second, is greater than the figure of their left, such as How many ascending numbers are between 4000 and 5000? (A) 7 (B) 8 (C) 9 (D) 10 (E) point problems In the equality KANG = AROO the letters denote the digits of four-digit numbers (different letters denote different digits). What is the value of the expression K A + N + G A + R O O? (A) 6 (B) 5 (C) 0 (D) 13 (E) All 4-digit positive integers with the same four digits as in the number 2013 are written in one line without gaps and in an increasing order. How many times does the pattern 2013 appear in this sequence of digits? (A) 1 (B) 2 (C) 3 (D) 4 (E) 9 Junior How many positive integers are multiple of 2013 and have exactly 2013 divisors (including 1 and the number itself)? (A) 0 (B) 1 (C) 3 (D) 6 (E) other Which of the following is the binary representation of the number 213? (A) (B) (C) (D) (E) What is the number of all different paths going from the point A to the point B at the 102

103 Junior, 4 point problems given graph? (A) 6 (B) 8 (C) 9 (D) 12 (E) Which of the following is a possible correct definition of π? (A) the ratio of the areas of a unit circle and a unit square (B) the circumference of the unit circle (C) the ratio of the diameter of a circle and its radius (D) the area of a circle with unit diameter (E) the ratio of the circumference of a circle and its radius How many three-digit numbers abc have the property that a is divided by b and b is divided by c? (A) 36 (B) 40 (C) 42 (D) 44 (E) Consider the regular hexagon in the picture, where P and Q are the middle points of sides AB and F E. What is the value of this relation A(AP QF )/A(BCDP )? (A) 5/8 (B) 1/2 (C) 3/4 (D) 1 (E) 3/ Five consecutive positive integers have the following property: there is three of them with the same sum than the sum of other two. How many such set of integers exist? (A) 0 (B) 1 (C) 2 (D) 3 (E) more than 3 Junior The solution of 2 3 x x = 8100 is (A) 1 (B) 2 (C) 3 (D) 4 (E) None of the previous Gauss wrote 10 consecutive integers on the board. He then added the 10 numbers and found that their sum was 150. For which of the following we can be certain that it is true? (A) The smallest integer he wrote on the board was

104 Junior, 4 point problems (B) The smallest integer he wrote on the board was 11 (C) The largest integer he wrote on the board was 19 or more (D) The largest integer he wrote on the board was 21 (E) Gauss made a mistake in this calculation If AA + BB + CC = 198 how much is A + B + C? (A) 8 (B) 9 (C) 18 (D) 19 (E) None of the previous How many of the following four numbers a) , b) , c) and d) 2012 ( ) 61 are multiples of 2013? (A) none (B) one (C) two (D) three (E) all four In triangle ABC the bisectrix of angle A and angle B intersect in point I. If C = γ, then AIB = (A) 2γ (B) 180 γ (C) 360 4γ (D) 60 + γ (E) γ Albert, Ben and Chris together take 4 lumps of sugar in their coffee. One of the possible distributions is: Albert 1, Ben 3 and Chris 0. How many distributions are there possible? (A) 4 (B) 6 (C) 9 (D) 12 (E) 15 Junior Somebody has a iron plate of 4 6 metre. He wants to make a closed, square box. He cuts off pieces from the plate and bent what is left over. Finally he closes the box by welding. How many metres is the weld seam? (A) 8 (B) 10 (C) 15 (D) 16 (E) that depends upon the dimensions of the box 104

105 Junior, 4 point problems ABCD is a rectangle; AB = 6 and BC = 3. The circles with midpoints A and B passing through C resp. D intersect in S. What is the distance from S to the segment CD? (A) 2 2 (B) 3 (C) π (D) 10 (E) A path has white and black tiles in a regular order. Ann walks along the path, from the left to the right. At each tile she mentions the number of white and black tiles she has passed already. w is the number of white tiles, b is the number of black tiles. Consider five statements: sometimes w = b + 2; if w = b, then after one step w > b; always w b; always w b; if w = b, then after one step w < b. How many of these statements are true? (A) 0 (B) 1 (C) 2 (D) 3 (E) In a 6x8-grid half of the squares are not intersected by one of the diagonals, as you can check in the picture. How many of the squares in a 7x10-grid are not intersected? (A) 26 (B) 30 (C) 35 (D) 40 (E) 44 Junior Between the touristic villages A and B is a railway. Trains go from A to B (or the other way) in exactly 2,5 hours. The trains depart simultaneously from A and B and departures take place with regular intervals. Each time a train arrives at its destination a train in the opposite direction departs. During its travel a train meets exactly 29 trains in the opposite direction. How many minutes are there between two consecutive departures? (A) 3 (B) 4 (C) 5 (D) 6 (E) Different letters represent different digits, same letters represent the same digits. For 105

106 Junior, 4 point problems example, AABAB can denote the number If the sum of the numbers AAB, ABA and BAA is 444, what is the result of A B? (A) 0 (B) 2 (C) 3 (D) 4 (E) 0 or We use black and white circles to build triangles such as in the picture. Every odd-numbered layer consists of black and white circles, alternating, starting with a black one. Every even-numbered layer is completely white. How many white circles does the smallest such triangle with 21 black circles have? (A) 34 (B) 45 (C) 57 (D) 66 (E) 70 Junior The picture shows five isosceles triangles with top angles 24, 48, 72, 96 and 120 the first multiples of the smallest top angle. All top angles have an integer number of degrees. We want to make a similar picture with as many isosceles triangles as possible. How many degrees is the smallest top angle in that case? (A) 3 (B) 6 (C) 8 (D) 9 (E)

107 Junior, 4 point problems The picture shows five isosceles triangles with top angles 24, 48, 72, 96 and 120 : the first multiples of the smallest top angle. All top angles have an integer number of degrees. We want to make a similar picture with as many isosceles triangles as possible. How many isosceles triangles do we need? (A) 3 (B) 5 (C) 8 (D) 10 (E) In every circle in the picture you must write one of the digits 1, 2, 3 or 4 in such a way that on every line you get each of the four digits. Two digits are already put down. Which digit must be put in the circle with the question mark? (A) 1 (B) 2 (C) 1 or 2 (D) 3 or 4 (E) impossible The edges of rectangle ABCD are parallel to the coordinate-axes. ABCD lies below the x-axis and on the left of the y-axis (i.e. in the third quadrant), see the figure. The coordinates of the four points A, B, C and D are all integers. We calculate for each of these points the number y-coordinate x-coordinate. Which of the four points gives the smallest number? Junior 107

108 Junior, 4 point problems (A) A (B) B (C) C (D) D (E) depends on the rectangle Using the diagonals John has painted four triangles in a square. After that he made an extra border of breadth 1 cm everywhere. As a result each triangle has the same area as the border. How many cm is the length of the original square? (A) 4 (B) (C) 8 (D) (E) Four people live in a house. A mail-man brings three letters to that house. What is the probability that no person in the house gets more than one letter? (A) 2/3 (B) 1/2 (C) 1/3 (D) 1/4 (E) 1/5 Junior ABC is a right-angled triangle, there the hypotenuse AB is constant and the altitude CD is as long as possible. Then the angle is (A) 15 (B) 30 (C) 45 (D) 60 (E) more information is needed The diagram shows three squares, two of which are formed by joining the midpoints of the sides of a larger square. The area of the smallest square is 4 cm 2. What is the sum of the areas of the three squares? (A) 16 cm 2 (B) 18 cm 2 (C) 24 cm 2 (D) 28 cm 2 (E) 36 cm 2 108

109 Junior, 4 point problems In the diagram, the shaded triangle connects the midpoints of three sides of a regular hexagon. What fraction of the hexagon is shaded? (A) 1 3 (B) 3 8 (C) 5 12 (D) (E) If a + 2b + 3c = 3 and a 2 + 4b 2 + 9c 2 = 53, what is the value of 2ab + 6bc + 3ac? (A) 44 (B) 22 (C) 22 (D) 44 (E) none of the previous The figure below shows zigzag of six squares of 1cm 1cm. Its perimeter is 14 cm. What is the perimeter of a similar zigzag consisting of the 2013 same squares? (A) 2022 (B) 4028 (C) 4032 (D) 6038 (E) The herd consists of young bulls and heifers. Heifers in the herd make up 55% of its total, and their total weight is 45% of the total weight of the herd. How many times more is the average weight of the bulls than the average weight of heifers? (A) (B) 3 2 (C) (D) 11 9 (E) On an infinite strip two machines every second place marks starting from the edge of the strip: the first machine - every 16 cm in red, the second machine - every 25 cm in blue. How long does it take for the red and blue tags to appear first at a distance of 1 cm from each other? (A) 7 s (B) 11 s (C) 17 s (D) 36 s (E) another answer Given 2n integers. What greatest number of pairwise sums of these numbers are odd numbers? (A) 1 2 n (n + 1) (B) n (n + 1) (C) 2n (n + 1) (D) n2 (E) 2n In an isosceles triangle ABC (AB = BC) on the side BC we have taken points K and M (K closer to B, than M) so that KM = AM and corners MAC and KAB are equal. What is the angle BAM? (A) 30 (B) 45 (C) 60 (D) 90 (E) can not be determined Junior The price of skis that cost 400 EUR got 25 % higher before the season. After the season the price decreased by 25 %. How much did they cost after the discount? (A) 500 EUR (B) 475 EUR (C) 400 EUR (D) 375 EUR (E) 350 EUR 109

110 Junior, 4 point problems The cleaning company charges 1,40 EUR for washing 1 m 2 of window glazing on both sides. The company MAX has 32 identical windows in their office. The dimensions of one of them are in the picture. How much did the company pay for having all the windows washed? (A) 44,80 EUR (B) 51,07 EUR (C) 72,96 EUR (D) 102,14 EUR (E) 145,92 EUR The visitors of cinema have to wait in the line for approximately 15 minutes if there are 3 cashiers open. If 2 more cashiers open, by how many minutes does the waiting time decrease? (A) 3 (B) 5 (C) 6 (D) 7 (E) Which of the following is not equal to 2013? (A) (B) (C) (D) (E) If a = 2b, c = a + 45 and a + 3b + 2c = 675 then a + b + c = (A) 65 (B) 130 (C) 175 (D) 195 (E) dwarfs are standing in a circle. Snow White put a cap on the head 3 of them. If a dwarf has a cap, his neighbours do not have. How many possibilities are there? (A) 7 (B) 14 (C) 21 (D) 28 (E) The average number of players in 8 teams that participated in the competition, was 6. Once there appeared the ninth team, the average number of players was equal to 7. How many players are there in the ninth team? (A) 6 (B) 7 (C) 8 (D) 9 (E) 15 Junior Find the greatest common divisor of all four-digit numbers written with numbers 3, 4, 5, 6. (A) 1 (B) 2 (C) 3 (D) 9 (E) In the squares of the 8 8 chessboard integers are written so that the numbers in adjacent squares differ by 1. We also know that numbers 3 and 17 are written there. How many different numbers are put in the table? (A) 10 (B) 12 (C) 15 (D) 18 (E) There are ten balls of three colors: blue, green and yellow. It is known that there are exactly 360 different ways to put them in a row. What may be the largest number of blue 110

111 Junior, 4 point problems balls? (A) 4 (B) 5 (C) 6 (D) 7 (E) How many scalene triangles are there with integer side lengths and with perimeter less than 11? (A) 0 (B) 1 (C) 6 (D) 7 (E) A class of students had a test. If each boy had got 3 points more for the test, then the average result of the class would had been 1,2 points higher than now. How many percent of the students of the class are girls? (A) 20% (B) 30% (C) 40% (D) 60% (E) it is impossible to determine The segment AB connects two opposit vertexes of a regular hexagon. The segment CD connects the midpoints of two opposite sides. Find product of AB and CD if the area of the hexagon is 60. (A) 40 (B) 50 (C) 60 (D) 80 (E) Working by turns for 9 hours four workers constructed a fence. Each worker worked so much time, how many it would be necessary for other three workers (working together) to construct exactly half of fence. For what time the fence would be constructed if all four workers worked together? (A) 1,5 h (B) 2 h (C) 2,5 h (D) 3 h (E) 4 h In the given figure CAD = 12, COD = 36, OC=10 cm, where O is a center of the circle. Find AB. (A) 8 cm (B) 9 cm (C) 10 cm (D) 12 cm (E) 14 cm Find ABC if A, B and C are midpoints of edges of a cube in following figure. Junior (A) 75 (B) 90 (C) 105 (D) 120 (E) Annika texts and calls her friends almost every day. Now she needs a new mobile phone and has to decide on a contract. The first contract has a basic fee of 10 Euros a month, 5 Cents 111

112 Junior, 4 point problems per minute and 6 Cents per text message. The second contract has no basic fee but each minute costs 10 Cents and each text message 9 Cents. In which case is the first contract cheaper? (A) 22 texts, 3 hours talk per month (C) 130 texts, 2 hours talk per month (E) 222 texts, 1 hour talk per month (B) 71 texts, 2.5 hours talk per month (D) 185 texts, 1.5 hours talk per month When Tanja and Arif found that 2 4 = 4 2, they tried to find other numbers with this property. Trying 3 and 9 they realized that 3 9 is bigger than 9 3. How many times bigger is it? (A) 2 times (B) 3 times (C) 9 times (D) 27 times (E) 39 times Joan wanted to draw two equilateral triangles attached to get a rhombus. But he did not hit correctly all the distances and, once he had done, Joanna was measured with a high precision instrument four angles and saw that they were not equal. Which of the five segments of the figure is the longest? (A) AB (B) AC (C) AD (D) BC (E) BD The number of bacteria in my Petri dish increases by 20% each day. How many days does it take before the number of bacteria is more than twice the initial number? (A) 3 (B) 4 (C) 5 (D) 6 (E) Dave cycles at a steady 12m/s and pursues Boris who cycles at a stately 3m/s. They are initially 60km apart. Roughly how long does it take for Dave to catch up with Boris? (A) 2 minutes (B) 2 hours (C) half a day (D) a day (E) 2 days The four-digit number xyxy is the square root of What is x + y? (A) 2 (B) 4 (C) 6 (D) 9 (E) 10 Junior A right circular cone has base radius 3 and height 4. What is its total surface area (including the base)? (A) 9π (B) 16π (C) 24π (D) 25π (E) 40π What is the value of x for which ( ) x 27 6 = 0.75? 64 (A) 1 (B) 2 (C) 3 (D) 4 (E) Fred is 15 years older than Ted. The product of their ages now is twice what it was 15 years ago. What is the sum of their current ages? (A) 35 (B) 45 (C) 55 (D) 75 (E)

113 Junior, 4 point problems Points A and D of the x-axis present the roots of the equation x 2 + ax + b = 0 and points B and C are the roots of the equation x 2 + cx + d = 0. The points A, B, C, D go from left to right and AB = CD. Then necessarily (A) a < c, b = d (B) a = c, b < d (C) a > c, b > d (D) a < c, b > d (E) a = c, b = d How many real numbers can satisfy the equation x a x b x c x d = 0? (A) 0 (B) 1 (C) 2 (D) 3 (E) Which of the straight lines (A) - (E) has the following property: for every point on it, the doubled sum of its x-coordinate and the tripled y-coordinate equals 6? (A) (B) (C) (D) (E) Let a 1 = 1 1+1, a 2 = , a 3 =,and so on. Find a (A) (B) (C) (D) (E) From a list of three numbers we call changesum the procedure to made a new list replacing each number by the sum of the other two. For example, from {3, 4, 6} changesum gives {10, 9, 7} and a new changesum leads to {16, 17, 19}. If we begin with the list {1, 2, 3}, How many consecutive changesums will be required to appear in the list the number 2013? (A) 8 (B) 9 (C) 10 (D) 2013 will appear several times (E) Never appear From a list of three numbers we call changediff the procedure to made a new list replacing each number by the absolute value of the difference of the other two. For example, from {3, 4, 6} changesum gives {2, 3, 1} and a new changesum leads to {2, 1, 1}. If we begin with the list {2013, 671, 33}, what is the maximum of the numbers of the list after 2013 consecutive changediffs? (A) 0 (B) 1 (C) 3 (D) 11 (E) another result Junior In a referendum scrutiny, when it had counted 30% of the total votes cast, count gives 33% of favorable votes. The it processes a new set of votes to reach 40% of the total of votes cast. If in this set the percentage of yes is 45%, what is the percentage of favorable votes in all 40% of votes counted? (A) 35% (B) 36% (C) 40% (D) 42% (E) 45% 113

114 Junior, 5 point problems What will be the units digit of the resulting number when we calculate the expression ? (A) 3 (B) 4 (C) 6 (D) 9 (E) We have many white cubes and many black cubes, all equal in terms of size. We want to build a cuboid composed exactly with 2013 of these cubes so that they are placed alternately a white cube and a black cube in all directions. If we start putting a black cube in one of the eight corners of the cuboid, how many black cubes have been used in construction? (A) 1006 (B) 1007 (C) 1012 (D) 1037 (E) It is impossible to construct a cuboid as the statement point problems Four points that lie inside a rectangle are connected by six line segments. The length d of the shortest segment is an integer. How many different integer values can the number d take? (A) 24 (B) 25 (C) 26 (D) 27 (E) After Snow-White wrote a positive integer n on the blackboard the dwarfs rewrote it as a product of seven (not necessarily different) prime multiples. The sum of the multiples did not exceed 17. Snow-White decreased each of the multiples by 1 and obtained the new product m. Then she noted that m divides n. Which of the following cannot be the sum of digits of n/m? (A) 8 (B) 9 (C) 10 (D) 11 (E) Calculating the sum , little Ben missed some of the summands and obtained a wrong sum divisible by Calculating the sum S = , little Ann missed exactly the same summands and obtained a wrong sum T divisible by Then T/S = (A) 2/3 (B) 670/671 (C) 1008/2013 (D) 6/11 (E) 2012/ The line P Q divides the trapezoid into two quadrangles with equal areas. What is the ratio x y? Junior (A) 3 2 (B) 4 3 (C) 5 4 (D) 5 3 (E) The rectangle shown in the figure below is once rotated around the x-axis resulting in the solid R x and once around the y-axis resulting in the solid R y. It s also given that a > b. Which of the following statements is true? 114

115 Junior, 5 point problems (A) R x = R y (B) R x R y but Vol(R x ) = Vol(R y ) (C) Vol(R x ) > Vol(R y ) (D) Vol(R x ) < Vol(R y ) (E) You cannot say any of the above in general as it depends on the exact values of a and b In equilateral triangle ABC, one knows the distances from an interior point M to each vertex of the triangle: AM = 5, BM = 3, CM = 4. What is the measure of angle BMC? (A) 90 (B) 100 (C) 120 (D) 135 (E) How many different sets A have the property that A {1, 2, 3, 4, 5} = {1, 2,..., 9}? (A) 512 (B) 256 (C) 255 (D) 32 (E) Consider the points A(1, 1), B(5, 1), C(2, 3), D(4, 3) and M(a, b) in the Cartesian coordinate system so that MA + MB + MC + MD has a minimum value. What is the value of the sum a + b? (A) 16/3 (B) 3 (C) 5/2 (D) 25/12 (E) 37/ How many factorial expressions n!(n + 1)! have exactly 51 zeros at the end? (A) 25 (B) 5 (C) 4 (D) 2 (E) If P (x) 3P (1 x) = 4(x 1), x real number, then what is the value of this expression P (1) + P (2) P (2012)? (A) (B) (C) (D) (E) Consider a truncated pyramid built out of a cardboard with the shape as shown in the picture (in which the square ABCD has a side equal to 1 and the trapeziums have angles of 60 ). Find its volume. Junior (A) 1/8 (B) 5 3/18 (C) 7 3/18 (D) 11 2/24 (E) 7 2/6 115

116 Junior, 5 point problems How many three-digit numbers abc have the property that a + b c? (A) 981 (B) 875 (C) 891 (D) 855 (E) The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 are written around a circle in arbitrary order. We add all numbers with their neighbours, we get 10 sums that way. What is the maximum value of the smallest from this sums? (A) 14 (B) 15 (C) 16 (D) 17 (E) A 1 A 2 B 2 B 1 is a quadrangle such that the lines A 1 A 2 and B 2 B 1 intersect at the left side and A 1 B 1 and A 2 B 2 intersect above. M 1 is the middle of A 1 B 1, M 2 is the middle of A 2 B 2. S is the point of intersection of the lines A 1 A 2 and M 1 M 2, T is the point of intersection of the lines B 1 B 2 en M 1 M 2. What is true? (A) S = T (B) S is at the left side of T (C) T is at the left side of S (D) S does not exist, because A 1 A 2 and M 1 M 2 are parallel (E) That depends upon the precise dimensions We use the number x to compute three other numbers: a = 2+x, b = 2 x and c = 2 x. Which of those three is the smallest and which is the biggest, depends on x. But the order a < b < c occurs for no number x. Which other order also occurs for no number x? (A) a < c < b (B) b < a < c (C) b < c < a (D) c < a < b (E) c < b < a Someone multiplies to numbers, both consisting of two equal digits. The product is So: aa bb = What is the sum of a and b? (A) 8 (B) 9 (C) 10 (D) 11 (E) 12 Junior Anna, Bill and Cathy go either to the Library or to the Gym. It is known that a) Either Ann or Bill go to the Library, but not both. b) If Ann goes to the Library then Bill goes to the Gym. c) Bill and Cathy do not both go to the Gym. Who could have gone to the Library last week and to the Gym this week? (A) Ann (B) Bill (C) Cathy (D) the situation described is impossible (E) we do not have enough information to decide What is the remainder of (1! + 2! + 3! !) 2 when divided by 5? (A) 0 (B) 1 (C) 2 (D) 3 (E) The diagram shows a rectangle, which is divided into four pieces by cutting along the 116

117 Junior, 5 point problems straight lines shown. The four pieces are then rearranged to form a square. What is the length of the perimeter of the square? (A) 40 (B) 48 (C) 52 (D) 56 (E) Uncle has bought all his nephews a gift consisting of a sweet, an orange, a cake, a bar of chocolate and a book. If he had bought only the sweets with the same money, they would have been 224, if only oranges they would have been 112, if only cakes - 56, if only bars of chocolate - 32, and if only books How many nephews has uncle got? (A) 5 (B) 6 (C) 7 (D) 8 (E) Yurko was walking down the street when saw a tractor that was pulling a long pipe. Deciding to measure its length, Yurko walked along the pipe against the movement of the tractor and counted 20 steps. Then he walked along the pipe with the movement of the tractor and counted 140 steps. Knowing that his step equals 1 m, Yurko was able to find the length of the pipe. What is it? (A) 30 m (B) 35 m (C) 40 m (D) 48 m (E) 80 m On 22 cards there have been written positive integers from 1 to 22. With these cards 11 fractions have been made. What is the greatest number of these fractions that can have integer values? (A) 7 (B) 8 (C) 9 (D) 10 (E) Romeo made a Valentine s surprise for his girlfriend. He cooked a dinner and laid the table beautifully. He even bought square double-coloured napkins (white on the one side, red on the other side) and folded each one of them as you can see in the picture. In what ratio is red and white colour of the folded napkin? (A) 1/2 (B) 1/3 (C) 2/3 (D) 8/11 (E) 8/13 Junior Only 10 girls had been in Shannon elementary school s volleyball team until yesterday. The couch found out their average height was 166 cm. Two more players came yesterday - Hannah and Michelle. Little Hannah is 10 cm shorter than Michelle. With arrival of these girls to the team, the average height decreased to 164 cm. How tall is Hannah? (A) 144 (B) 149 (C) 154 (D) 158 (E)

118 Junior, 5 point problems Andrew wants to make a polystyrene notice board in the shape of a flower for his sister Jane. The petals of the flower will be formed by 6 identical half-circles. The middle circle will have the radius of 5 dm. How large (in rounded dm2) will the area of the notice board be? (A) 75 dm 2 (B) 124 dm 2 (C) 137 dm 2 (D) 143 dm 2 (E) 156 dm How many such triangles are there, which vertecies are from the vertecies of a regular polygon with 13 sides and the centre of the outside circle of the polygon is inside of the triangle? (A) 72 (B) 85 (C) 91 (D) 100 (E) other value Norbert draw 10 triangles on the plane. Maximum how many part are there? (A) 264 (B) 266 (C) 268 (D) 270 (E) What is the smallest integer which has not less than 12 positive divisors? (A) 2 11 (B) (C) (D) (E) In the right-angled triangle the altitude, dropped to hypotenuse, divides it into segments, the difference of which is equal to the smaller leg. Find the ratio of the triangle legs (smaller to larger). (A) 1 : 3 (B) 1 : 2 (C) 2 : 3 (D) 1 : 3 (E) 2 : 5 Junior From three-digit number the sum of cubes of its digits has been substracted. What is the greatest result to get? (A) 292 (B) 326 (C) 375 (D) 396 (E) A car left point A and drove along the straight road at a speed of 50 km/h. Then every hour a car left point A, and each next car was 1 km/h faster than the previous one. The last car (at a speed of 100 km/h) left 50 hours after the first one. What is the speed of the car which was in front of all the column 100 hours later after the start of the first car? (A) 50 km/h (B) 66 km/h (C) 75 km/h (D) 84 km/h (E) 100 km/h students took part in the Math Olympiad, 50 students participated in the Physics Olympiad, 48 students were involved in the Computer Sciences Olympiad. When each student was asked how many competitions he participated in, the answer in at least two was given two times less than the answer not less than one and the answer in three was given three times 118

119 Junior, 5 point problems less than the answer not less than one. How many students participated in the competitions? (A) 198 (B) 150 (C) 124 (D) 108 (E) On each face of the cube there is an integer supplied. At each vertex of the cube there is the product of the numbers which are put on the adjacent sides of this vertex. The sum of all numbers in the vertices is equal to What can the sum of all the numbers on the faces be equal to? (A) 1003 (B) 151 (C) 103 (D) 91 (E) The big square is made of small 1002 squares of these squares had side with length 1 cm. The area of the big square couldn t be equal to: (A) 45 2 (B) (C) 75 2 (D) 51 2 (E) In the squares of the 8 8 chessboard integers are written so that the numbers in adjacent squares differ by 1. We also know that numbers 3 and 17 are written there. Find the sum of all the numbers noted in the table. (A) 640 (B) 400 (C) 320 (D) 150 (E) Into which greatest number of different rectangles can checked square 7 7 be cut (without residue) along the lines of the grid? (A) 6 (B) 8 (C) 9 (D) 10 (E) Three painters should paint 100 identical cubes. Each side of a cube should be painted completely only by one painter. Different painters can paint different sides of the same cube, but no two painters can paint the same cube simultaneously. For painting of any side of a cube 10 seconds are required to each painter. To postpone one cube and to pass to painting other cube (or, maybe, to two painters to exchange cubes) also 1 second is required. Painters begin to work simultaneously. For what least time painters can paint completely all 100 cubes? (A) 2000 seconds (B) 2033 seconds (C) 2034 seconds (D) 2040 seconds (E) 2053 seconds How many natural numbers with five digits have the last three digits all equal, are multiples of four but are not divisible by eight? (A) None (B) 45 (C) 90 (D) 100 (E) XY Z is an equilateral triangle with the radius of the circumscribed circle equal to 1. M and N are respectively the midpoints of segments XY and XZ. P is one of the intersection Junior points of the line MN and the circumscribed circle (see picture). length of segment NP? (A) 15 3 (B) 1 (C) 3 (D) 3 3 (E) What is the Let (a n ) n N be a sequence such that a n 0 ( n N) and lim n a 2n+1 a 2n = c. For which 119

120 Junior, 5 point problems of the following values of c the number a = 0, a 1, a 2,..., a n,... is certainly rational? (A) 1 (B) 2 (C) 3 (D) 4 (E) How many operations must make in order to receive from 2 number 1025? (A) 20 (B) 24 (C) 25 (D) 30 (E) The series 2, 3, 5, 6, 7, 10,... contains the consecutive positive integers which are not perfect squares or perfect cubes either. Which number is placed 2013-rd? (A) 2064 (B) 2065 (C) 2066 (D) 2067 (E) In school marathon 101 runners took part. It has appeared on the finish, that Bobby outstripped 2 times more participants, than outstripped Michael, and Michael has outstripped 3 times more participants, than have outstripped the Bobby. What place Michael occupied (i.e. what under the account he was on finish)? (A) 20 (B) 21 (C) 40 (D) 41 (E) Each member of the serries 1, 2, 4, 8, 16, 23, 28, 38,... is obtained by adding to the previous one the sum of its digits. Which number is not a member of the series? (A) 2006 (B) 2013 (C) 2014 (D) 2021 (E) Using the digits 1, 3, 5, 7 and 9 Johnny made three various five-digit numbers (in each number each of these digits used exactly once). To what of the following values the sum of these three numbers cannot be equal? (A) (B) (C) (D) (E) Peter wrote down all integers from 1 up to 11. Then he deleted one of these numbers, and marked each of four verticies and six edges of a tetrahedron with ten remained numbers using each of them exactly once. It appeared that for any two verticies of the tetrahedron the sum of two numbers which mark these verticies equals the number which mark the edge connecting these two verticies. What number Peter has deleted? (A) 1 (B) 6 (C) 8 (D) 10 (E) 11 Junior What the maximal quantity of numbers it is possible to arrange in a row in order to the sum of any of five successively going numbers be positive, and the sum of any seven successively going numbers be negative? (A) 8 (B) 10 (C) 12 (D) 14 (E) it is impossible to do John has wrote down 10 integers. Then he calculated all possible pairwise products of these numbers. It appeared that exactly 16 products are positive. How many products can be negative? (A) 16 (B) 20 (C) 25 (D) 29 (E) impossible to determine 120

121 Junior, 5 point problems trees (oaks and birches) grow along a road. The number of trees between any two oaks does not equal 5. What the greatest number of oaks can be among these 100 trees? (A) 48 (B) 50 (C) 52 (D) 60 (E) the situation is not possible In each cell of a 5 5 board one kangaroo sits on. At midday every kangaroo jumps over the side of a cell on the next cell. What greatest number of cells can appear empty? (A) 12 (B) 14 (C) 15 (D) 16 (E) Find the greatest value, which can be received if one is allowed to place brackets somehow in expression (A) 9 (B) 15 (C) 39 (D) 45 (E) other answer In the figure asterisk stands for digits. Find the sum of the factors. (A) 346 (B) 356 (C) 446 (D) 456 (E) other answer There are few straight lines on the plane. The number of intersections with other these lines is written near each line. Number 3 is written near line a, number 4 is written near line b. Some number not equal 3 and nor 4 is written near line c. Find this number. (A) 2 (B) 5 (C) 6 (D) 7 (E) other answer Seven people attend the first lesson in driving school. Among them there is one person that is friends with exactly 3 of the others and there are 5 people that are friends with exactly 2 of the others. How many friends does the fifth person have among the others? (A) 1, 2 or 3 (B) 1, 3 or 5 (C) 2, 3 or 5 (D) 1, 4 or 5 (E) 2, 4 or Marc got a brand new bike for his birthday. It has 15 gears. The three gear wheels in front have 22, 32 and 44 teeth and the five in the back have 11, 15, 16, 30 and 32 teeth. If Marc pedals with constant speed, with how many different speeds can he go (i. e. how many different speed transformations does his new bike have)? (A) 15 (B) 14 (C) 13 (D) 12 (E) Let ABCD be a rectangle, P the midpoint of side AB and Q the point on P D such that CQ is perpendicular to P D. Then for sure (A) CQ = DQ (B) BC = BQ (C) BC = CQ (D) BQ = DQ (E) BQ = CQ Junior The total height of a container made of a cylinder and a cone is 9 units. If you spill liquid that fills a third of the container with the cone located at the bottom, then the liquid reaches a height of 5 units. If we reverse the container, which will be the height of liquid? (Note: in all 121

122 Junior, 5 point problems cases we assume the container placed vertically) (A) 1,5 units (B) 2 units (C) 2,5 units (D) 3 units (E) It depends on the radius of cylinder and cone How many different pairs of integers are there such that their product equals 5 times their sum? (A) 4 (B) 5 (C) 6 (D) 7 (E) A number N is the product of three distinct primes. How many distinct factors does N have? (A) 8 (B) 16 (C) 27 (D) 63 (E) The integers 1, 3, 8 and n, where n < 1000, have the property that the product of any two of them is one less than a square number. What is the sum of the digits of n? (A) 3 (B) 6 (C) 8 (D) 9 (E) On a day when Sundollars Coffee Shop sells 477 cups of coffee to 190 customers and every customer has at least one cup of coffee, what is the maximum possible value of the median number of cups of coffee bought per customer? (A) 1.5 (B) 2 (C) 2.5 (D) 3 (E) Through a point A inside a tetrahedron we draw the planes parallel to its four faces. What is the number of parts into which these planes cut the tetrahedron? (A) 12 (B) 13 (C) 14 (D) 15 (E) 16 Junior 122

123 Student, 3 point problems 6 Student point problems The sum of all the integers that divide number 12 is equal to (A) 28 (B) 27 (C) 16 (D) 6 (E) The two diagonals e and f of a quadrilateral form a right angle at their intersection point: e f. The lengths of e and f are 12 cm and 30 cm, respectively. What is the area of the quadrilateral? (A) 90 cm 2 (B) 180 cm 2 (C) 240 cm 2 (D) 360 cm 2 (E) Missing data In triangle ABC, the measure of angle BAC is 120, AM AB and BM = MC. What is the true relation? (A) AB = 2AC (B) AC = 2AB (C) BC = 2AB (D) BC = 2AC (E) AB = AC Consider the square in the picture with side equal to 1. M is the middle point of side BC and N is the middle point of segment BM. Find the area of the surface hatched in the picture. (A) 1/3 (B) 1/7 (C) 1/21 (D) 2/21 (E) 1/ How many natural numbers n have the property that n/(2012 n) is still a nonnegative integer number? (A) 1 (B) 2 (C) 3 (D) 4 (E) Calculate the value of (A) 11 (B) 13 (C) (D) (E) None of the previous Which of the following numbers is not a divisor of ? (A) 51 (B) 91 (C) 105 (D) 225 (E) all the previous are divisors n is an integer satisfying : n + (n + 1) + (n + 2) (n + 20) < What is the Student 123

124 Student, 3 point problems maximum value of n? (A) 85 (B) 86 (C) 87 (D) 89 (E) Let f: R R be the function defined by : f is periodic with period 5; the restriction of f to [ 2, 3[ is x f (x) = x 2. What is f (2013)? (A) 0 (B) 1 (C) 2 (D) 4 (E) If a 3 a 11 a n = a 2013, a > 0, then n = (A) 61 (B) 1980 (C) 1999 (D) 2000 (E) Let f be a linear function for which f(2013) f(2001) = 100. What is f(2031) f(2013)? (A) 75 (B) 100 (C) 120 (D) 125 (E) When I found that 2 4 = 4 2, I tried to find other numbers with this property. Trying 2 and 8 I realized that 2 8 is bigger than 8 2. How many times bigger is it? (A) 2 times (B) 4 times (C) 8 times (D) 16 times (E) 32 times Which of the following numbers is the largest? (A) (B) (C) (D) (E) Given that 2 < x < 3 how many of the following statements are true? 4 < x 2 < 9 4 < 2x < 9 6 < 3x < 9 0 < x 2 2x < 3 (A) 0 (B) 1 (C) 2 (D) 3 (E) For how many integer values of n is n 2 + 2n prime? (A) 0 (B) 1 (C) 2 (D) 3 (E) an infinite number Which of the options is closer to the value of x y if x = y = ? (A) 1 (B) 2 (C) 3 (D) 4 (E) If f(x) = 1 and g(x) = 1 x, what is the value of gfgfgf gf(2013), if g (and also x f) is applied 100 times? (A) (B) (C) 2014 (D) 2012 (E) 1 Student In the triangles of the picture, numbers 1, 2, 3, 4, 5, 6 must be written in such a way that each 6 triangles forming a hexagon have different numbers (notice that some triangles belong to more than one hexagon). Some of the numbers have already been written. What 124

125 Student, 3 point problems number should be written in the shaded triangle? (A) 1 (B) 2 (C) 4 (D) 5 (E) In the coordinate plane, the square whose vertices are (0, 0), (1, 0), (1, 1) and (0, 1) has area 1. All the vertices of the convex polygon P have coordinate integers and the area of P is 5. Let n be the number of vertices in the interior of P. Which of the following statements is true? (A) n can be any integer between 0 and 4 (B) n has to be 0 (C) n must be even (D) there are exactly 4 possibilities for n (E) n < Let ABCDEF and KLMNOP be regular hexagons, K is the midpoint of the segment AB. If AB = 2 cm, then KL is: (A) 3 2 cm (B) 1 cm (C) 3 cm (D) 2 3 cm (E) cm If x 1 and x 2 are the solutions of the equation 2 3 x x = 3, then how many of the following expressions are true: x 1 + x 2 < 27, x 1 + x 2 > 27, x 1 x 2 < 3, x 1 x 2 > 3? (A) 1 (B) 2 (C) 3 (D) 4 (E) If x 0 is the solution of the equation log 3 (x + 1) + log 3 (x + 3) 1 = 0, then (A) x 0 [ 4, 1) (B) x 0 [ 1, 6) (C) x 0 (0, 7) (D) the equation doesn t have solution (E) there are two good answers, B and C Four circles of radius 1 are touching each other and a smaller circle as seen in the picture. Student 125

126 Student, 3 point problems What is the radius of the smaller circle? (A) 2 1 (B) 1 2 (C) 3 4 (D) 3 4 (E) A prism has 2013 faces in total. How many edges has the prism? (A) 2013 (B) 2011 (C) 4022 (D) 6033 (E) Impossible Marta works in a medicine s factory. She should weigh a substance packed in three plastic bags. She can t weigh all three together, and neither can weigh one by one because the balance only works for weights between 100 g and 200 g. Then, she weighs two bags at a time. The first bag and the second weigh 175 g together, the first and the third weigh 155 g together and the two last bags weigh 150 g together. How much does the second bag weigh? (A) 95 (B) 90 (C) 65 (D) 85 (E) The volume of a rectangular parallelepiped is 2013 and its dimensions are integers greater than one. Which can be the area of one of its faces? (A) 34 (B) 182 (C) 671 (D) 66 (E) Consecutive integers are added, and the sum is in each case. If the quantity of summands is smaller than 10, how many cases are there? (A) 1 (B) 2 (C) 3 (D) 5 (E) None of these Emi draws the figure shown aside in the grid. How many trapeziums does Emi find? (A) 6 (B) 8 (C) 10 (D) 12 (E) 7 Student Consider the set of all numbers {n = a b} with the property that a is a three-digit number and b the number that results from a by writing the digits in reverse order. Which of the following primes is a common divisor of all numbers n in the set? (A) 2 (B) 5 (C) 7 (D) 11 (E) When frozen water (i.e. ice) melts, its volume increases by 1. By how much does its

127 Student, 4 point problems volume decrease when it freezes again? (A) 1 10 (B) 1 11 (C) 1 12 (D) 1 13 (E) A three-dimensional object bounded only by polygons is called a polyhedron. What is the smallest number of polygons that can bound a polyhedron, if we know that one of them has 12 sides? (A) 12 (B) 13 (C) 16 (D) 18 (E) Six superheroes capture 20 villains. The first superhero captures one villain, the second captures two villains and the third captures three villains. The fourth superhero captures more villains than any of the other five. What is the smallest number of villains the fourth superhero must have captured? (A) 3 (B) 4 (C) 5 (D) 6 (E) The year 2013 has the property that its number is made up of the consecutive digits 0, 1, 2 and 3. How many years have passed since the last time a year was made up of four consecutive digits? (A) 467 (B) 527 (C) 581 (D) 693 (E) point problems Given a polynomial W (x) = (a x)(b x) 2, where a < b. The graph of the function y = W (x) is in one of the following figures. In which one? (A) (B) (C) (D) (E) How many rectangles, whose one of sides is equal to 5, have the property, that after cutting off a rectangle of area 4, at least one of the two rectangles obtained is a square? (A) 1 (B) 2 (C) 3 (D) 4 (E) If f(x) = x 2 + 1, then f(f(x)+x) f(x) is equal to (A) x 2 + x + 1 (B) x 2 + 2x + 2 (C) x 2 + x + 1 (D) x 2 + 2x + 1 (E) x 2 + x Student 127

128 Student, 4 point problems The curves of the following functions are either all or part of a curve of the given type. Not all are correct. How many are wrong? f 1 (x) = 1 + 2x + 3 parabola; f 2 (x) = 1 hyperbola; 2x 3 f 3 (x) = (1 x) parabola; f 4 (x) = (1 x 2 ) + 3 circle; f 5 (x) = 1 + e 2x 3 hyperbola; f 6 (x) = 1 (2x 3) line; f 7 (x) = hyperbola; x 2 (A) 1 (B) 2 (C) 3 (D) 4 (E) Which of the following numbers is the best approximation of the ratio of the circumference of a circle and its radius? (A) 0.32 (B) 1.57 (C) 2 (D) 3.14 (E) The sides DC and AB of a tetragon ABCD are divided by two lines into 3 equal segments. The sides DA and CB of the same tatragon are also divided by two lines into other 3 equal segments. What is the ration between the area of ABCD the area of the tetragon formed by the four lines (S ABCD /S MNP Q =?) (A) 3 (B) 4 (C) 6 (D) 9 (E) If 3P (x) + P ( x) = 4 sin x cos x, x real number, then P (x) =? (A) sin x (B) cos x (C) sin x cos x (D) sin 2x (E) cos 2x In order to get a number divided by 11, Maria must substract a number equal to her age from number How old is Maria? (A) 10 (B) 9 (C) 8 (D) 7 (E) What is the value of x for which x is divisible by 1000? (A) 192 (B) 208 (C) 321 (D) 581 (E) Iulian has written an algoritm in order to create a sequence of numbers as a 1 = 1, a m+n = a m + a n + mn, where m and n are natural numbers. Find the value of the a 100 (A) 100 (B) 1000 (C) 2012 (D) 4950 (E) Vlad has drawn the graph of a function f: R R (see figure). How many real solutions does the following equation have f(f(f(x))) = 0? Student 128

129 Student, 4 point problems (A) 4 (B) 3 (C) 2 (D) 1 (E) Radu has identical plastic pieces which are in the shape of a regular pentagon. Find the minimum number of pieces he needs to place in order to get a ring. (A) 8 (B) 9 (C) 10 (D) 12 (E) How many integers n exist such that both n 3 and 3n are three figure numbers? (A) 33 (B) 34 (C) 300 (D) 333 (E) no such numbers exist What is the remainder of when divided by ? (A) 0 (B) 1 (C) 2 (D) (E) In a group of 20 kangaroo families there is a total of 63 children. If each family has 3 or 4 or 5 children, how many of these families have 4 children? (A) 1 (B) 2 (C) 3 (D) 4 (E) none of the previous Ignaz has a regular octahedron and eight regular tetrahedra, all with congruent faces. He glues the tetrahedra onto the octahedron such that the triangular faces cover each other completely. Which of the following best describes the resulting polyhedron? (A) convex polyhedron with 24 triangular faces (B) concave polyhedron with 24 triangular faces (C) regular tetrahedron (D) regular octahedron (E) regular icosahedron Let O be the center of the circle circumscribing the triangle ABC. Which angles can Student 129

130 Student, 4 point problems be in this triangle, if the quadrilateral ABOC is a rhomb? (A) 60, 60, 60 (B) 45, 45, 90 (C) 30, 60, 90 (D) 30, 30, 120 (E) 15, 15, The sequences is defined as a n = 2 n n + 1 for all integer n 1. Find a 1 + a a 9. (A) 945 (B) 986 (C) 1000 (D) 1068 (E) A box contains 900 cards numbered from 100 to 999, two different cards have different numbers. Franois draws some cards, without replacement, and, for each one, he forms the sum of the number s figures. How many cards, at least, should he draw to be certain to have three cards giving the same sum? (A) 51 (B) 52 (C) 53 (D) 54 (E) Let f : N N 0 be the function defined by f(n) = n n 1 if n is even, f(n) = if n is odd, 2 2 for all natural number n. For k a positive integer f k (n) denotes de number represented by the expression f(f(...f(n)...))), where the symbol f appears k times. The number of solutions of the equation f 2013 (n) = 1 is (A) 0 (B) 4026 (C) (D) (E) an infinity If x is a real number such that 4x i= 4 (i + 4) = 0, then x is equal to (A) 1352 (B) 1365 (C) 1456 (D) 1470 (E) In the triangle ABC the points M and N from the side AB are such that AN = AC and BM = BC. Find ACB if MCN = 43. (A) 86 (B) 89 (C) 90 (D) 92 (E) Some function y=f(x) is defined on the set of all real numbers and is monotonously increasing. Then the function y = (f(x)) 2 (A) decreases (C) has no more than one maximum (E) is monotonic (B) increases (D) has no more than one minimum Let a, b, c be mutually distinct positive integers and f(x) function defined by f(x) = (x a)(x b) (a c)(b c) (x b)(x c) (x c)(x a) + + (b a)(c a) (c b)(a b). Student The value of f(2012 f(2013)) is (A) ab + bc + ca (B) 1 (C) (D) 1 (E) Another number 130

131 Student, 4 point problems Which of the following rules defines the same function on the real numbers as f(x) = x? (A) g 1 (x) = x 2 (B) g 2 (x) = x2 (C) g 3 (x) = x2 x x x 1 (D) g 4 (x) = x3 + x (E) g x 2 5 (x) = ( x ) Starting with a square of side length 1 Anna constructs alternatingly squares and circles as shown in the picture. What is the radius of the 10th circle? (A) 10 (B) 16 (C) 8 2 (D) 32 (E) The figure shows the graphic of a polynomial function such as p(x) = ax 4 +bx 3 +cx 2 +d. (A) p(0) Which of the following numbers is the smallest? (C) The minimum of p in [ 2, 1] (E) lim x 4 p(x) (B) The sum of the coefficients of p (D) p (1) p ( 1) For x 8, how many integer values of x satisfy the equation x 3 (x 1) 3 = p where p is a prime number? (A) 6 (B) 7 (C) 8 (D) 9 (E) Given that xy x = 9 and y + yx = 16, what is the value of y x? (A) 7 (B) 1 (C) 1 (D) 7 (E) The sum of the first n positive integers is a three-digit number in which all of the digits are the same. What is the sum of the digits of n? (A) 6 (B) 9 (C) 12 (D) 15 (E) How many different pairs of integers are there such that their product equals 5 times their sum? (A) 4 (B) 5 (C) 6 (D) 7 (E) A number N is the product of three distinct primes. How many distinct factors does N 2 have? (A) 8 (B) 16 (C) 27 (D) 63 (E) 64 Student 131

132 Student, 5 point problems A circular carpet is placed on a floor of square tiles. All the tiles which have more than one point in common with the carpet are marked grey. Which of the following is an impossible outcome? (A) (B) (C) (D) (E) How many prime numbers are of the form n 2 + 6n 27, where n is an integer? (A) 0 (B) 1 (C) 2 (D) 3 (E) more than In a referendum scrutiny, when it had counted 30% of the total votes cast, count gives 33% of favorable votes. Later, when the count has reached 40% of the total votes cast, the percentage of favorable votes increased to 36%. What is the percentage of favorable votes counted between these two moments of the scrutiny? (A) 45% (B) 72% (C) 69% (D) 34.5% (E) It is impossible to happen Consider the following proposition about a function f on the set of integer numbers: For any even x, f(x) is even. What would be the negation of this proposition? (A) For any even x, f(x) is odd (B) For any odd x, f(x) is even (C) For any odd x, f(x) is odd (D) There is a even number x such that f(x) is odd (E) There is an odd number x such that f(x) is odd We have many white cubes and many black cubes, all equal in terms of size. We want to build a cuboid composed exactly with 2013 of these cubes so that they are placed alternately a white cube and a black cube in all directions. If we start putting a black cube in one of the eight corners of the cuboid, how many black squares will we see at the exterior surface of the cuboid? (A) 887 (B) 888 (C) 890 (D) 892 (E) It depends on the dimensions of the cuboid We have a sum of three three-digit integers resulting If all nine digits are different, what is the digit that not appear? (A) 2 (B) 3 (C) 4 (D) 7 (E) 0 Student point problems Four points that lie inside a rectangle are connected by six line segments. The length d of the shortest segment is an integer. What is the biggest possible value of d? (A) 24 (B) 25 (C) 26 (D) 27 (E)

133 Student, 5 point problems On a wooden cube little Tom drew some line segments connecting opposite vertices of its sides - one segment on each side. Which of the following cannot be the total number of triangles formed by the drawn segments? (A) 0 (B) 1 (C) 2 (D) 3 (E) After Snow-White wrote a positive integer n on the blackboard the dwarfs rewrote it as a product of seven (not necessarily different) prime multiples. Snow-White increased each of the multiples by 1 and obtained the new product m. Then she noted that n divides m. Which of the following can be the sum of digits of m/n? (A) 3 (B) 4 (C) 6 (D) 8 (E) Let n > 3 be a positive integer. Calculating the sum (n + 1), little Ben missed some of the summands and obtained a wrong sum divisible by n. Calculating the sum (n + 2), little Ann missed exactly the same summands and obtained a wrong sum divisible by n + 3. Which of the following could be n? (A) 20 (B) 21 (C) 201 (D) 2013 (E) The town is inhabited by knights who always say the truth and liars who always lie. Once the inspector came to the town. He asked every inhabitant one question about some other person in the town if that person was a liar. He never asked about the same person repeatedly. Then he arrested all who were told to be liars and left the town with the arrested. All remaining knights whose answer caused the arrest got very upset and left the town as well. There were thrice less of them than those knights who were arrested. What part of all the people who disappeared from the town did the knights constitute? (A) 1/3 (B) 2/3 (C) 3/5 (D) 4/7 (E) 8/ How many solutions (x, y), where x and y are real numbers, does the equation x 2 +y 2 = x + y have? (A) 1 (B) 5 (C) 8 (D) 9 (E) Infinitely many Define function f as follows Then (A) f is increasing. (B) f is decreasing. f(x) = x 3 1, x R. (C) f is increasing on (, 1] and decreasing on [1, ). (D) f is decreasing on (, 1] and increasing on [1, ). (E) The definition of f given above contains an error All the balls with the radius equal to 1cm need to be packed in 30cm 30cm 20cm boxes. What is the maximum number of balls packed in a such box so that the box lid closes completely? (A) 225 (B) 232 (C) 450 (D) 247 (E) 240 Student 133

134 Student, 5 point problems Grandma has ten grandchildren whose ages form the set {1, 4, 7, 10, 13, 16, 19, 22, 25, 28}. Everytime three of them come to visit her, she makes a cake on which she puts as many candles as the sum of their ages. How many different sums can grandma get if any three of them come to visit her? (A) 13 (B) 21 (C) 22 (D) 24 (E) A bowl is filled with 100 C water, and another bowl is half-filled with 0 C water. Pour water from the first bowl into the second bowl until this is completely filled. After stirring, pour water from the second bowl into the first bowl until this is completeley filled and so on. What is the temperature of the water in the filled bowl after such 10 steps? (A) 50 (B) 60 (C) 63 (D) 67 (E) What is the sum of all two-digit numbers that have the property of being divided by each of its digits? (A) 135 (B) 485 (C) 495 (D) 530 (E) The first day of each year, Mark and Liza put in a bag as many beans as the number of the year (for instance, 2013 this year) and play in the following way. One of them, randomly chosen, starts taking out of the bag a number of beans at his choice, at least 1 and no more than 8. Than the other does the same and so on. The one who takes out the last bean from the bag is the winner. Which is the next year in which the one who starts playing as the second one can be sure to be the winner, due to some winning strategy? (A) 2015 (B) 2016 (C) 2017 (D) 2018 (E) The diagram shows two squares of equal side length placed so that they overlap. The squares touch at a common vertex and the sides make an angle of 45 with each other. What is the area of the overlap as a fraction of the area of one square? (A) 1 2 (B) 1 2 (C) (D) 2 1 (E) How many solutions in positive integers has the equation x 2 y 3 = 6 12? (A) 6 (B) 7 (C) 8 (D) 9 (E) a and b are positive integers, a 4 + a 5 + a 6 + b 6 + b 7 + b 8 + b 9 = 2013, a + b =? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 Student There are 100 points on the plane, there is not 3 collinear of them. We connect points with segments. The segments do not cross eachother. Maximum how many segments is possible to draw this way? (A) 288 (B) 291 (C) 294 (D) 297 (E) boys are standing in a circle. Anna put a cap on the head 4 of them. If a boy has a 134

135 Student, 5 point problems cap, his neighbours do not have. How many possibilities are there? (A) 70 (B) 105 (C) 120 (D) 126 (E) What is the smallest integer which has more than 12 positive divisors? (A) 2 12 (B) (C) (D) (E) Which digit is in 2013-th place in the line of where each integer N is written in turn N times? (A) 4 (B) 5 (C) 6 (D) 7 (E) There are 2013 points marked inside the square. Some of them are connected to the vertices of the square and with each other so that the square has been broken into triangles. All marked points are the vertices of these triangles. The number of triangles formed is: (A) 2013 (B) 2015 (C) 4026 (D) 4028 (E) impossible to determine The digits from 1 to 9 are placed in the 9 cells of a 3 times 3 square, each digit - once. Consider the sums of the numbers in each of the shown four 2 times 2 squares. What is the greatest possible value of the least sum? Squares.pdf (A) 21 (B) 22 (C) 23 (D) 24 (E) What the maximal quantity of numbers it is possible to arrange in a row in order to the sum of any of seven successively going numbers be negative, and the sum of any nine successively going numbers be positive? (A) 10 (B) 12 (C) 14 (D) 16 (E) it is impossible to do There are few straight lines on the plane. The number of intersections with other these lines is written near each line. Number 3 is written near line a, number 4 is written near line b. Some number not equal 3 and nor 4 is written near line c. Find the number of lines on the plane. (A) 4 (B) 5 (C) 6 (D) 7 (E) other answer In the following sequence of integers each element a n is obtained from the previous one a n 1 in the following way: If a n 1 is an integer, then a n = a n If not, then a n = a n 1 5. The sequence starts with a 1 = 123. What is a 2013, the 2013th element in the sequence? (A) 2 (B) 1 (C) 0 (D) 1 (E) The roundabout shown in the picture is entered by 5 cars at the same time, each one from a different direction. Each of the cars drives less than one round and no two cars leave the roundabout in the same direction. How many different combinations are there for the cars leaving the roundabout? Student 135

136 Student, 5 point problems (A) 24 (B) 44 (C) 60 (D) 81 (E) On the island of Knights and Knaves there lives only two types of people: Knights (who always spkeak the truth) and Knaves (who always lie). I met two men who lived there and asked the taller man if they were both Knights. He replied, but I could not figure out what they were, so I asked the shorter man if the taller was a Knight. He replied, and after that I knew which type they were. Were the men knights or knaves? (A) They were both Knights. (B) They were both Knaves. (C) The taller was a Knight and the shorter was a Knave. (D) The taller was a Knave and the shorter was a Knight. (E) Not enough information is given The cube in the figure is cut by a plane passing through vertices D, E and B, so that the vertex A is separated from the center of the cube. Then all the other vertices are cut off as well by similar plane cuts. What will the piece containing the center of the cube look like? Student (A) (B) (C) 136

137 Student, 5 point problems (D) (E) From a list of three numbers we call changediff the procedure to made a new list replacing each number by the absolute value of the difference of the other two. For example, from {3, 4, 6} changesum gives {2, 3, 1} and a new changesum leads to {2, 1, 1}. If we begin with the list {a, b, c}, where 0 < a < b < c < 2013 what is, certainly, the maximum of the numbers of the list after 2013 consecutive changediffs? (A) 0 (B) 1 (C) minimum of {a, b, c} (D) great common divisor of {a, b, c} (E) we can not predict without knowing {a, b, c} Student 137