Kangourou sans Frontières

Transcription

1 Kangourou sans Frontières Selected problems for KSF contest 2014

2 KSF 2014 selected problems PreEcolier 3 points # 1 (2451). Ladybird will sit on a flower that has five petals and three leaves. On which of the following flowers will ladybird sit? (A) (B) (C) (D) (E) # 2 (2854). In what order do you meet the shapes starting from the arrow? 1

3 KSF 2014 selected problems PreEcolier (A),, (B),, (C),, (D),, (E),, # 3 (2564). How many more grey squares than white ones can you see? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10 # 4 (3312). Put the animials in line from the smallest to the largest. What animal is in the middle? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 # 5 (2563). Ann has twelve of these tiles. She makes one line with the design. Ann starts at the left side. How does the line end? (A) (B) (C) (D) (E) 2

4 KSF 2014 selected problems PreEcolier # 6 (3277). Which is the shadow of the girl? (A) (B) (C) (D) (E) # 7 (2856). A square was composed of 25 small squares, but some of these small squares are lost. How many are lost? (A) 6 (B) 7 (C) 8 (D) 10 (E) 12 # 8 (3279). How many ducks balance the crocodile? 3

5 KSF 2014 selected problems PreEcolier (A) (B) (C) (D) (E) 4 points # 9 (3526). When the ant goes from home following these arrows: 3, 3, 3, 1, it comes to the ladybird Which animal would it come to, if it goes from home following these arrows: 2, 2, 3, 3, 2, 2? (A) (B) (C) (D) (E) # 10 (3607). The kangaroo is inside how many circles? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 # 11 (2865). A square was cut into 4 parts as shown in the picture. Which of the following shapes 4

6 KSF 2014 selected problems PreEcolier cannot be made with these 4 parts? (A) (B) (C) (D) (E) # 12 (2569). Which form fits exactly the one given above? (A) (B) (C) (D) (E) # 13 (2867). Walking from K to O along the lines pick up the letters KANGAROO in the correct order. What is the lenght of the shortest walk in meters? 1 m 1 m (A) 16 m (B) 17 m (C) 18 m (D) 19 m (E) 20 m # 14 (3433). How many numbers are greater than 10 and less than or equal to 31 which can be written with digits 1, 2 or 3 only? You can repeat digits. (A) 2 (B) 4 (C) 6 (D) 7 (E) 8 5

7 KSF 2014 selected problems PreEcolier # 15 (2570). Seven sticks lie on top of each other. Stick 2 is at the bottom. Stick 6 is at the top. Which stick is in the middle? (A) 1 (B) 3 (C) 4 (D) 5 (E) 7 # 16 (3311). How many frogs did the three pelicans catch? (A) 1 (B) 2 (C) 4 (D) 9 (E) 12 5 points # 17 (3307). The chess board is damaged. How many black squares on the right side of the line are missing? (A) 11 (B) 12 (C) 13 (D) 14 (E) 15 # 18 (3398). Rabbit Venya eats cabbages and carrots. Each day he eats either 10 carrots, or 2 6

8 KSF 2014 selected problems PreEcolier cabbages. Last week Venya ate 6 cabbages. How many carrots did he eat? (A) 20 (B) 30 (C) 34 (D) 40 (E) 50 # 19 (2455). What should you put in the square to get a correct diagram? (A) 38 (B) : 8 (C) 45 (D) 6 (E) : 6 # 20 (2684). Put the digits 2, 3, 4 and 5 in the squares and calculate the sum to get the largest value. What is that value? + (A) 68 (B) 77 (C) 86 (D) 95 (E) 97 # 21 (2870). The central cell of the square was removed. We cut it into equal pieces. Which piece is not possible to get? (A) (B) (C) (D) (E) # 22 (2539). To get the product of 2 x 3 x 15, Bill has to press the keys of his calculator seven times: Bill wants to multiply all numbers from 3 to 21, using his calculator. At least, how many times will he press the keys of his calculator? (A) 19 (B) 31 (C) 37 (D) 50 (E) 60 # 23 (2879). Fedya has 4 red cubes, 3 blue cubes, 2 green cubes and 1 yellow cube. He builds a tower (see the picture) in such a way that no two adjacent cubes have the same colour. What is the? colour of the middle cube? (A) red (B) blue (C) green (D) yellow 7

9 KSF 2014 selected problems PreEcolier (E) impossible to determine # 24 (2571). Cogwheel A turns round completely once. At which place is x now? (A) a (B) b (C) c (D) d (E) e 8

10 KSF 2014 selected problems Ecolier 3 points # 1 (3062). Which drawing is the central part of the picture with the star? (A) (B) (C) (D) (E) # 2 (3340). Jacky wants to insert the digit 3 somewhere in the number Where should she insert the digit 3 if she wants her five-digit number to be as small as possible? (A) in front of 2014 (B) between the 2 and the 0 (C) between the 0 and the 1 (D) between the 1 and the 4 (E) behind 2014 # 3 (3284). Which houses are made using exactly the same pieces of triangular or rectangular shape? New diagram required. (A) 1, 4 (B) 3, 4 (C) 1, 4, 5 (D) 3, 4, 5 (E) 1, 2, 4, 5 # 4 (3048). When Koko the Koala does not sleep, he eats 50 grams of leaves per hour. Yesterday, he slept 20 hours. How many grams of leaves did he eat yesterday? (A) 0 (B) 50 (C) 100 (D) 200 (E) 400 # 5 (2578). Maria subtracts and gets as results the numbers from zero to five. She connects the dots, starting at the dot with the result 0 and ending at the dot with the result 5. Which figure does she get? (A) (B) (C) (D) (E) # 6 (3493). Adam built fewer sandcastles than Martin but more than Susan. Lucy built more sandcastles than Adam and more than Martin. Dana built more sandcastles than Martin but fewer 1

11 KSF 2014 selected problems Ecolier than Lucy. Who of them built the most sandcastles? (A) Martin (B) Adam (C) Susan (D) Dana (E) Lucy # 7 (2574). Monica writes numbers in the diagram so that each number is the product of the two numbers below it. Which number should she write in the grey cell? (A) 0 (B) 1 (C) 2 (D) 4 (E) 8 # 8 (2580). Ann has four pieces as shown. With these pieces she can completely cover the shape. Where should she put the piece? (A) (B) (C) (D) (E) 4 points # 9 (3189). Mr. Brown has painted flowers on the store window (see picture). How does thess flowers look like from the other side of the window? 2

12 KSF 2014 selected problems Ecolier (A) (B) (C) (D) (E) # 10 (3115). Some candies were in a bowl. Sally took half of the candies. Then Tom took half of the candies left in the bowl. After that Clara took half of the remaining candies. In the end there were 6 candies in the bowl. How many candies were in the bowl at the beginning? (A) 12 (B) 18 (C) 20 (D) 24 (E) 48 # 11 (3157). Which tile must be added in the picture so that the white area is as big as the black area?? (A) (B) (C) (D) (E) # 12 (3004). Paula shoots arrows at the following target. When she misses, she obtains zero points. Paula shoots two arrows and adds the number of points. Which of the following sums cannot be her score? (A) 60 (B) 70 (C) 80 (D) 90 (E) 100 # 13 (2529). Mary had equal number of grey, black, and striped tokens. She used some of these 3

13 KSF 2014 selected problems Ecolier tokens to make a pile. You can see all used tokens in the figure. She still has five tokens which are not on the pile. How many black tokens did she have at the beginning? New diagram required. (A) 5 (B) 6 (C) 7 (D) 15 (E) 18 # 14 (3399). Rabbit Borya likes cabbages and carrots very much. In a day he eats either 9 carrots, or 2 cabbages, or 1 cabbage and 4 carrots. During one week Borya has eaten 30 carrots. How many cabbages has he eaten during this week? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10 # 15 (3498). The solid in the picture was made by sticking eight equal cubes together. How does this solid look from above? (A) (B) (C) (D) (E) # 16 (2582). How many dots are there in this picture? (A) 180 (B) 181 (C) 182 (D) 183 (E) points 4

14 KSF 2014 selected problems Ecolier # 17 (3360). On Kangaroo planet each kangyear has 20 kangmonths and each kangmonth has 6 kangweeks. How many kangweeks are there in one quarter of a kangyear? (A) 9 (B) 30 (C) 60 (D) 90 (E) 120 # 18 (2586). Seven children are standing in a circle. No two boys are standing next to each other. No three girls are standing next to each other. Which of these is true for the number of girls standing in the circle? (A) only 3 is possible (B) 3 and 4 are possible (C) only 4 is possible (D) 4 and 5 are possible (E) only 5 is possible # 19 (3652). Eve arranged cards in a line as it is shown in the figure below. At each move Eve is allowed to interchange any two cards. What is the smallest number of moves Eve needs to get the word KANGAROO? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 # 20 (2575). A sequence of triangles of diamonds is made. The first three stages are shown. In each stage a line of diamonds is added. In the bottom line the outer diamonds are white. All other diamonds in the triangle are black. How many black diamonds has the figure in stage 6? (A) 19 (B) 21 (C) 26 (D) 28 (E) 34 # 21 (3316). Kangaroo Hamish bought toys and gave the shop-assistant 150 Kangcoins. He received 20 Kangcoins back. Then he changed his mind and exchanged one of the toys for another. He got back an additional 5 Kangcoins. What toys did Hamish leave the store with? 5

15 KSF 2014 selected problems Ecolier New diagram required. (A) the carriage and the plane (C) the carriage and the tram (E) the bus, the motorcycle and the tram (B) the carriage and the bus (D) the motorcycle and the tram # 22 (2754). Write each of the numbers 0, 1, 2, 3, 4, 5, 6 in the squares to make the addition correct. Which digit will be in the grey square? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 + # 23 (3108). What is the largest number of small squares which can be shaded so that no square made of four shaded small squares appears on the figure? (A) 18 (B) 19 (C) 20 (D) 21 (E) 22 # 24 (3402). Nick has written each of the numbers from 1 to 9 in the cells of the 3 3 table. Only four of these numbers can be seen in the figure. Nick has noticed that for the number 5 the sum of the numbers in the neighbouring cells equals 13 (neighbouring cells are cells sharing sides). He noticed the same applies for the number 6. Which number has Nick written in the shaded cell? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 6

16 Æ Ã KSF 2014 selected problems Benjamin 3 points # 1 (3005). Arno spelled the word KANGAROO with cards showing one letter at a time. Unfortunately some cards were tipped. Tipping back twice he can correct the letter K and tipping once he can correct the A - see the figures. How many times does he need to tilt for all of the letters to be correct? Ã Ê Ã Ã (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 # 2 (3554). A cake weights 900 g. Paul cuts it in 4 pieces. The biggest piece is as heavy as the 3 others weight altogether. What s the weight of the biggest piece? (A) 250 g (B) 300 g (C) 400 g (D) 450 g (E) 600 g # 3 (2592). Two great rings, one grey, one white, are linked in each other. Peter, in front of the rings, sees the rings as in the picture. Paul is behind the rings. What does he see? (A) (B) (C) (D) (E) # 4 (3319). In the following addition, some of the digits have been replaced by stars = 309 What is the sum of the missing digits? (A) 0 (B) 1 (C) 2 (D) 3 (E) 10 # 5 (3042). What is the difference between the smallest 5-digit number and the largest 4-digit number? (A) 1 (B) 10 (C) 1111 (D) 9000 (E) 9900 # 6 (3385). A square of perimeter 48 cm is cut into 2 pieces to make a recangle (see picture). What is the perimeter of the rectangle?if we cut a rectangle in half and place one piece above the other we create a square whose area is 144 cm 2. What is the perimeter of the rectangle? 1

17 KSF 2014 selected problems Benjamin (A) 24 cm (B) 30 cm (C) 48 cm (D) 60 cm (E) 72 cm # 7 (3006). Katrin has 38 matches. She builds a triangle and a square, using all the matches. Each side of the triangle consists of 6 matches. How many matches are in each side of the square? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 # 8 (3160). The pearl necklace in the picture contains dark grey pearls and shiny white pearls. Arno wants to have 5 of the dark grey pearls. He can only take pearls from either end of the necklace, and so he has to take some of the white pearls also. What is the smallest number of white pearls Arno has to take? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 # 9 (2496). Harry participated in a broom flight contest which consisted of 5 laps. The times when Harry passed the starting point are shown in the picture. Which lap took the shortest time? (A) the first (B) the second (C) the third (D) the fourth (E) the fifth # 10 (3205). Ben s digital watch is not working properly. The three horizontal lines in the rightmost digit on the watch are not displayed. Ben is looking at his watch and the time has just changed from the one shown on the left to the one shown on the right. What time is it now? (A) 12:40 (B) 12:42 (C) 12:44 (D) 12:47 (E) 12:49 4 points # 11 (3161). Which tile must be added to the picture so that the white area is as large as the black area?? (A) (B) (C) (D) (E) It is impossible. 2

18 KSF 2014 selected problems Benjamin # 12 (3394). Henry and John started walking from the same point. Henry went 1 km north, 2 km west, 4 km south and finally 1 km west. John went 1 km east, 4 km south and 4 km west. Which of the following must be the final part of his walk in order to reach the same point as Henry? (A) He has already reached the same point. (B) 1 km north. (C) 1 km north-west. (D) More than 1 km north-west. (E) 1 km west. # 13 (3114). At the summer camp, 7 pupils eat ice cream every day, 9 pupils eat ice cream every second day and the rest of the pupils don t eat ice cream at all. Yesterday, 13 pupils had ice cream. How many pupils will eat ice cream today? (A) 7 (B) 8 (C) 9 (D) 10 (E) it cannot be determined # 14 (2399). Kangaroos A, B, C, D and E are sitting in that order, clockwise, around a circular table. Exactly when the bell rings, each kangaroo but one exchanges its position with a neighbour. The resulting positions, clockwise and starting with A, are A, E, B, D, C. Which kangaroo did not move? (A) A (B) B (C) C (D) D (E) E # 15 (3057). A square can be formed using four of these five pieces. Which one will not be used? A B C D E (A) A (B) B (C) C (D) D (E) E # 16 (2492). A natural number has three digits. When we multiply the digits we get 135. What result do we get if we add the digits? (A) 14 (B) 15 (C) 16 (D) 17 (E) 18 # 17 (3357). In a restaurant there are 16 tables, each having either 3, 4 or 6 chairs. Together, the tables having 3 or 4 chairs can accommodate 36 people. Knowing that the restaurant can accommodate 72 people, how many tables are there with 3 chairs? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 # 18 (2759). The points A, B, C, D, E, F are on a straight line in that order. We know that AF = 35, AC = 12, BD = 11, CE = 12 and DF = 16. What is the distance BE? (A) 13 (B) 14 (C) 15 (D) 16 (E) 17 # 19 (3595). Parisa set her stones in groups on the desk. After she arranged the stones in groups of 3, she found that there were 2 stones left. Then she arranged the stones in groups of 5, and again 3

19 KSF 2014 selected problems Benjamin there were 2 stones left. At least how many more stones does she need so that there won t be any left when she arranges them in groups of 3 and in groups of 5? (A) 3 (B) 1 (C) 4 (D) 10 (E) 13 # 20 (3113). The faces of a cube are numbered 1, 2, 3, 4, 5, and 6. The faces 1 and 6 have a common edge. The same is true for faces 1 and 5, faces 1 and 2, faces 6 and 5, faces 6 and 4, and faces 6 and 2. Which number is on the face opposite the one with number 4? (A) 1 (B) 2 (C) 3 (D) 5 (E) it cannot be determined 5 points # 21 (2594). The cube in the picture is made of 27 small cubes. How many small cubes do you have to take away to see the following result when looking from the right, from above, and from the front? (A) 4 (B) 5 (C) 6 (D) 7 (E) 9 # 22 (3110). There are 5 songs: song A lasts 3 min, song B 2 min 30 s, song C 2 min, song D 1 min 30 s, and song E 4 min. These 5 songs are playing in the order A, B, C, D, E in a loop without any breaks. Song C was playing when Andy left home. He returned home exactly one hour later. Which song was playing when Andy got home? (A) A (B) B (C) C (D) D (E) E # 23 (3403). Dan entered the numbers 1 to 9 in the cells of a 3x3 table. He began by placing the numbers 1, 2, 3 and 4 as shown in the picture. It happened that for the number 5, the sum of the numbers in the adjacent cells (having a common side) is equal to 9. What is the sum of the numbers adjacent to the number 6? (A) 14 (B) 15 (C) 17 (D) 28 (E) 29 # 24 (3415). Trees grow on only side of Park Avenue. There are 60 trees in total. Every second tree is a maple, and every third tree is either a linden or a maple. The remaining trees are birches. How many birches are there? (A) 10 (B) 15 (C) 20 (D) 24 (E) 30 4

20 KSF 2014 selected problems Benjamin # 25 (3489). A thin colourful ribbon is stuck on a transparent plastic cube (see the picture). Which of the following pictures doesn t show the cube from any perspective? (A) (B) (C) (D) (E) # 26 (3096). The king and his messengers are travelling from the castle to the summer palace at a speed of 5 km/h. Every hour, the king sends a messenger back to the castle, who travels at a speed of 10 km/h. What is the time interval between any two consecutive messengers arriving at the castle? (A) 30 min (B) 60 min (C) 75 min (D) 90 min (E) 120 min # 27 (3594). There were 3 one-digit numbers on the blackboard. Ali added them up, and got 15. Then he erased one of the numbers and wrote the number 3 in its place. Then Reza multiplied the three numbers on the blackboard and got 36. What are the possibilities for the number that Ali erased? (A) either 6 or 7 (B) either 7 or 8 (C) only 6 (D) only 7 (E) only 8 # 28 (3400). Rabbit Vasya loves cabbages and carrots. In a day, he eats either 9 carrots, or 2 cabbages, or 1 cabbage and 4 carrots. But some days he only eats grass. Over the last 10 days, Vasya ate a total of 30 carrots and 9 cabbages. On how many of these 10 days did he eat only grass? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 # 29 (3635). In Fabuland, every sunny day is immediately preceded by two consecutive rainy days. Also, five days after any rainy day, it is another rainy day. It is sunny today. For how many days at most can we predict the weather with certainty? (A) 1 day (C) 4 days (E) We can predict the weather every day from here on (B) 2 days (D) We cannot predict even one day ahead # 30 (2602). Granny has 10 grandchildren. Alice is the eldest. One day, Granny notices that her grandchildren all have different ages. If the sum of her grandchildrens ages is 180, what is the youngest Alice could be? (A) 19 (B) 20 (C) 21 (D) 22 (E) 23 5

21 KSF 2014 selected problems Cadet 3 points # 1 (2778). Each year, the date of the Kangaroo competition is the third Thursday of March. What is the latest possible date of the competition in any year? (A) 14th March (B) 15th March (C) 20th March (D) 21st March (E) 22nd March # 2 (2729). How many quadrilaterals of any size are shown in the figure? New Diagram required (A) 0 (B) 1 (C) 2 (D) 4 (E) 5 # 3 (3555). What is the result of: : ? (A) 0 (B) 1 (C) 2013 (D) 2014 (E) 4028 # 4 (3379). The area of rectangle ABCD is 10. Points M and N are midpoints of the sides AD and BC. What is the area of quadrilateral MBND? New diagram required. (A) 0,5 (B) 5 (C) 2,5 (D) 7.5 (E) 10 # 5 (2620). The product of two numbers is 36 and their sum is 37. What is their difference? (A) 1 (B) 4 (C) 10 (D) 26 (E) 35 # 6 (2816). Wanda has several square pieces of paper of area 4. She cuts them into squares and right-angled triangles in the manner shown in the first diagram. She takes some of the pieces and makes the bird shown in the second diagram. What is the area of the bird? New diagram required (without the eye) (A) 3 (B) 4 (C) 9/2 (D) 5 (E) 6 1

22 KSF 2014 selected problems Cadet # 7 (3469). A bucket was half full. A cleaner added 2 litres to the bucket. The bucket was then three-quarters full. What is the capacity of the bucket? (A) 10 l (B) 8 l (C) 6 l (D) 4 l (E) 2 l # 8 (3012). Georg built the shape shown using seven unit cubes. How many such cubes does he have to add to make a cube with edges of length 3? (A) 12 (B) 14 (C) 16 (D) 18 (E) 20 # 9 (3128). Which of the following calculations gives the largest result? (A) (B) (C) (D) (E) # 10 (3165). The necklace in the picture contains grey beads and white beads. Arno takes one bead after another from the necklace. He always takes a bead from one of the ends. He stops as soon as he has taken the fifth grey bead. What is the largest number of white beads that Arno can take? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 4 points # 11 (2413). Jack has a piano lesson twice a week and Hannah has a piano lesson every other week. In a given term, Jack has 15 more lessons than Hannah. How many weeks long is their term? (A) 30 (B) 25 (C) 20 (D) 15 (E) 10 # 12 (3216). In the diagram, the area of each circle is 1cm 2. The area common to two overlapping circles is 1 8 cm2. What is the area of the region covered by the five circles? (A) 4cm 2 (B) 9 2 cm2 (C) 35 8 cm2 (D) 39 8 cm2 (E) 19 4 cm2 2

23 KSF 2014 selected problems Cadet # 13 (3296). This year a grandmother, her daughter and her granddaughter noticed that the sum of their ages is 100 years. Each of their ages is a power of 2. How old is the granddaughter? (A) 1 (B) 2 (C) 4 (D) 8 (E) 16 # 14 (2553). Five equal rectangles are placed inside a square with side 24 cm, as shown in the diagram. What is the area of one rectangle? (A) 12 cm 2 (B) 16 cm 2 (C) 18 cm 2 (D) 24 cm 2 (E) 32 cm 2 # 15 (3391). The heart and the arrow are in the positions shown in the figure. At the same time the heart and the arrow start moving. The arrow moves three places clockwise and the heart moves four places anticlockwise and then stop. They continue the same routine over and over again. After how many routines will the heart and the arrow land in the same triangular region for the first time? (A) 7 (B) 8 (C) 9 (D) 10 (E) It will never happen # 16 (3560). The diagram shows the triangle ABC in which BH is a perpendicular height and AD is the angle bisector at A. The obtuse angle between BH and AD is four times the angle DAB (see the diagram). What is the angle CAB? (A) 30 (B) 45 (C) 60 (D) 75 (E) 90 # 17 (3146). Six boys share a flat with two bathrooms which they use every morning beginning at 7:00 o clock. There is never more than one person in either bathroom at any one time They spend 8, 10, 12, 17, 21 and 22 minutes at a stretch in the bathroom respectively. What is the earliest time that they can finish using the bathrooms? (A) 7:45 (B) 7:46 (C) 7:47 (D) 7:48 (E) 7:50 # 18 (3550). A rectangle has sides of length 6 cm and 11 cm. One long side is selected. The bisectors of the angles at either end of that side are drawn. These bisectors divide the other long side 3

24 KSF 2014 selected problems Cadet into three parts. What are the lengths of these parts? (A) 1 cm, 9 cm, 1 cm (B) 2 cm, 7 cm, 2 cm (C) 3 cm, 5 cm, 3 cm (D) 4 cm, 3 cm, 4 cm (E) 5 cm, 1 cm, 5 cm # 19 (3414). Captain Sparrow and his pirate crew dug up several gold coins. They divide the coins amongst themselves so that each person gets the same number of coins. If there were four fewer pirates, then each person would get 10 more coins. However, if there were 50 fewer coins, then each person would get 5 fewer coins. How many coins dig they dig up? (A) 80 (B) 100 (C) 120 (D) 150 (E) 250 # 20 (2985). The average of two positive numbers is 30% less than one of them. By what percentage is the average greater than the other number? (A) 75% (B) 70% (C) 30% (D) 25% (E) 20% 5 points # 21 (3404). Andy enters all the digits from 1 to 9 in the cells of a 3x3 table, so that each cell contains one digit. He has already entered 1, 2, 3 and 4, as shown. Two numbers are considered to be neighbours if their cells share an edge. After entering all the numbers he notices that the sum of the neighbours of 9 is 15. What is the sum of the neighbours of 8? (A) 12 (B) 18 (C) 20 (D) 26 (E) 27 # 22 (3150). An antique scale is not working properly. If something is lighter than 1000 g, the scale shows the correct weight. However, if something is heavier than or equal to 1000 g, the scale can show any number above 1000 g. We have 5 weights A g, B g, C g, D g, E g each under 1000 g. When they are weighed in pairs, the scale shows the following: B + D = 1200, C + E = 2100, B + E = 800, B + C = 900, A + E = 700. Which of the weights is the heaviest? (A) A (B) B (C) C (D) D (E) E # 23 (2509). Quadrilateral ABCD has right angles only at vertices A and D. The numbers show the areas of two of the triangles. What is the area of ABCD? New diagram required. (A) 60 (B) 45 (C) 40 (D) 35 (E) 30 # 24 (2987). Liz and Mary compete in solving problems. Each of them is given the same list of 100 problems. For any problem, the first of them to solve it gets 4 points, while the second to solve it gets 1 point. Liz solved 60 problems, and Mary also solved 60 problems. Together, they got 312 4

25 KSF 2014 selected problems Cadet points. How many problems were solved by both of them? (A) 53 (B) 54 (C) 55 (D) 56 (E) 57 # 25 (3731). David rides his bicycle from Edinburgh to his croft. He was going to arrive at 15:00, but he spent 2/3 of the planned time covering 3/4 of the distance. After that, he rode more slowly and arrived exactly on time. What is the ratio of the speed for the first part of the journey to the speed for the second part? (A) 5 : 4 (B) 4 : 3 (C) 3 : 2 (D) 2 : 1 (E) 3 : 1 # 26 (3149). We have four identical cubes (see picture). They are arranged so that a big black circle appears on one face, as shown in the second picture. What can be seen on the opposite face? (A) (B) (C) (D) (E) # 27 (2766). A group of 25 people consists of knights, serfs and damsels. Each knight always tells the truth, each serf always lies, and each damsel alternates between telling the truth and lying. When each of them was asked: Are you a knight?, 17 of them said Yes. When each of them was then asked: Are you a damsel?, 12 of them said Yes. When each of them was then asked: Are you a serf?, 8 of them said Yes. How many knights are in the group? (A) 4 (B) 5 (C) 9 (D) 13 (E) 17 # 28 (3662). Several different positive integers are written on the board. Exactly two of them are divisible by 2 and exactly 13 of them are divisible by 13. Let M be the greatest of these numbers. What is the smallest possible value of M? (A) 169 (B) 260 (C) 273 (D) 299 (E) 325 # 29 (3148). On a pond there are 16 water lily leaves in a 4 by 4 pattern as shown. A frog sits on a leaf in one of the corners. It then jumps from one leaf to another either horizontally or vertically. The frog always jumps over at least one leaf and never lands on the same leaf twice. What is the greatest number of leaves (including the one it sits on) that the frog can reach? (A) 16 (B) 15 (C) 14 (D) 13 (E) 12 # 30 (3732). A 5 5 square is made from 1 1 tiles, all with the same pattern, as shown. Any two adjacent tiles have the same colour along the shared edge. The perimeter of the large square consists 5

26 KSF 2014 selected problems Cadet of black and white segments of length 1. What is the smallest possible number of such unit black segments? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 6

27 KANGAROO 2014, ANSWERS. PREECOLIER ECOLIER BENJAMIN CADET JUNIOR STUDENT 1 B 1 D 1 C 1 D 1 B 1 C 2 A 2 D 2 D 2 D 2 A 2 C 3 D 3 A 3 D 3 A 3 E 3 B 4 B 4 D 4 A 4 B 4 A 4 C 5 C 5 A 5 A 5 E 5 C 5 E 6 D 6 E 6 D 6 E 6 A 6 E 7 D 7 E 7 B 7 B 7 B 7 E 8 B 8 C 8 B 8 E 8 B 8 D 9 A 9 E 9 B 9 B 9 E 9 D 10 C 10 E 10 C 10 D 10 C 10 C 11 D 11 B 11 E 11 E 11 C 11 A 12 E 12 D 12 B 12 B 12 C 12 B 13 C 13 B 13 D 13 C 13 B 13 B 14 D 14 B 14 B 14 E 14 D 14 C 15 A 15 C 15 B 15 E 15 A 15 D 16 D 16 B 16 D 16 C 16 E 16 D 17 B 17 B 17 A 17 B 17 E 17 A 18 D 18 C 18 D 18 E 18 C 18 B 19 E 19 B 19 E 19 D 19 D 19 A 20 D 20 C 20 A 20 A 20 C 20 C 21 E 21 A 21 D 21 E 21 D 21 E 22 D 22 D 22 A 22 D 22 D 22 D 23 A 23 D 23 E 23 B 23 D 23 E 24 A 24 D 24 C 24 D 24 B 24 A 25 E 25 C 25 C 25 D 26 D 26 A 26 C 26 A 27 B 27 B 27 B 27 C 28 C 28 C 28 E 28 B 29 C 29 A 29 B 29 D 30 E 30 B 30 C 30 D