Kangaroo 2015 Past Papers Cadet Level. Muhammad Javed Iqbal

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1 Kangaroo 2015 Past Papers Cadet Level Muhammad Javed Iqbal

2 Kangaroo 2005 Cadet Cadet Max Time: 75 min 3-Point-Problems 1. There are eight kangaroos in the cells of the table (see the figure on the right). Any kangaroo can jump into any free cell. Find the least number of the kangaroos which have to jump into another cell so that exactly two kangaroos remain in any row and in any column of the table. (A) 1 (B) 2 (C) 3 (D) 4 2. How many hours are there in half of a third of a quarter of a day? (A) 3 1 (B) 2 1 (C) 1 (D) 2 3. We have a cube with edge 12 cm. The ant moves on the cube surface from point A to point B along the trajectory shown in the figure. Find the length of ant s path. (A) 48 cm (C) 60 cm (B) 54 cm (D) it is impossible to determine 4. At National School, 50% of the students have bicycles. Of the students who have bicycles, 30% have black bicycles. What percent of students of National School have black bicycles? (A) 15% (B) 20% (C) 25% (D) 40% 5. In triangle ABC, the angle at A is three times the size of that at B and half the size of the angle at C. What is the angle at A? (A) 30 (B) 36 (C) 54 (D) The diagram shows the ground plan of a room. The adjacent walls are perpendicular to each other. Letters a, b represent the dimensions (lengths) of the room. What is the area of the room? (A) 2ab + a(b a) (B) 3a(a + b) a 2 (C) 3a(b a) + a 2 (D) 3ab 1

3 Kangaroo 2005 Cadet Max Time: 75 min 7. Nida cuts a sheet of paper into 10 pieces. Then she took one piece and cut it again to 10 pieces. She went on cutting in the same way three more times. How many pieces of paper did she have after the last cutting? (A) 36 (B) 40 (C) 46 (D) Some crows are sitting on a number of poles in the back of the garden, one crow on each pole. For one crow there is unfortunately no pole. Sometime later the same crows are sitting in pairs on the poles. Now there is one pole without a crow. How many poles are there in the back of the garden? (A) 2 (B) 3 (C) 4 (D) 5 9. A cube with the side 5 consists of black and white unit cubes, so that any two adjacent (by faces) unit cubes have different color, the corner cubes being black. How many white unit cubes are used? (A) 62 (B)63 (C) 64 (D) To make concrete, mix 4 shovels of stone, 2 shovels of sand and 1 shovel of cement. The number of shovels of stone required to make 350 shovels of concrete is: (A) 200 (B) 150 (C) 100 (D) 50 4-Point-Problems 11. Edward has 2004 marbles. Half of them are blue, one quarter are red, and one sixth are green. How many marbles are of some other colour? (A) 167 (B) 334 (C) 501 (D) A group of friends is planning a trip. If each of them would make a contribution of Rs.14 for the expected travel expenses, they would be Rs.4 short. But if each of them would make a contribution of Rs.16, they would have Rs.6 more than they need. How much should each of the friends contribute so that they collect exactly the amount needed for the trip? (A) Rs (B) Rs (C) Rs (D) Rs In the diagram, the five circles have the same radii and touch as shown. The small square joins the centres of the four outer circles. The ratio of the area of the shaded part of all five circles to the area of the unshaded parts of all five circles is (A) 1 : 1 (B) 2 : 5 (C) 2 : 3 (D) 5 : 4 2

4 Kangaroo 2005 Cadet Max Time: 75 min 14. Some angles in quadrilateral ABCD are shown in the figure. If BC AD, then what is the angle ADC? (A) 75 (B) 50 (C) 55 (D) 65 B C 50? A 15. The watchman works 4 days a week and has a rest on the fifth day. He had been resting on Sunday and began working on Monday. After how many days will his rest fall on Sunday? D (A) 31 (B) 12 (C) 34 (D) Which of the following cubes has been folded from the plan on the right? (A) (B) (C) (D) 17. The diagram shows an equilateral triangle and a regular pentagon. What, in degrees, is the size of the angle marked x? (A) 108 (B) 120 (C) 132 (D) This is a products table. What two letters represents the same number? 7 J K L 56 M 36 8 N O 27 6 P 6 18 R S 42 (A) L and M (B) O and N (C) R and P (D) M and S 3

5 Kangaroo 2005 Cadet Max Time: 75 min 19. Mike chose a three-digit number and a two-digit number. Find the sum of these numbers if their difference equals 989. (A) 1000 (B) 1001 (C) 1009 (D) Point A lays on a circle with a center in point O. What a part of the circle filled the points, which are closer to O, than to A? (A) 3/4 (B) 2/3 (C) 1/2 (D) 5/6 5-Point-Problems 21. Two rectangles ABCD and DBEF are shown in the figure. What is the area of the rectangle DBEF? (A) 10 cm 2 (B) 12 cm 2 (C) 14 cm 2 (D) 16 cm For a natural number N, by its length we mean the number of factors in the representation of N as a product of prime numbers. For example, the length of the number 90 (when N=90) is 4 because 90 = How many odd numbers less than 100 have length 3? (A) 3 (B) 5 (C) 7 (D) non of these 23. A caterpillar starts from his home at 9:00 a.m. and move directly on a ground, turning after each hour at 90 to the left or to the right. In the first hour he moved 1 m, in the second hour 2 m, and so on. At what minimum distance from his home the caterpillar would be at 4:00 p.m. in the afternoon? (A) 0 m (B) 1 m (C) 1.5 m (D) 2.5 m 24. How many degrees are the sum of the 10 angles which you can see in the picture (A) 300 (B) 360 (C) 600 (D) The average of 10 different positive integers is 10. How much can be the biggest one among the 10 numbers at most? (A) 10 (B) 14 (C) 55 (D) 60 4

6 KANGAROO-2005: Correct Answers N Ecolier Benjamin Cadet Junior Student 1 A D A C C 2 A C C C C 3 B A C B B 4 B C A C D 5 A C C D B 6 A C D D A 7 A C C E E 8 B A B C C 9 A B A E B 10 A C A C A 11 A D A D D 12 A B C C A 13 B D C E E 14 B C D C D 15 A A C D A 16 B D D B A 17 B D C D D 18 B C D C C 19 B B C E D 20 B C B D B 21 B E E 22 B D A 23 A E C 24 D E C 25 C A B 26 D E 27 B A 28 E E 29 C D 30 E B

7 Kangaroo 2006 Cadet Max Time: 95 min Cadet: Class (7-8) 3-Point-Problems 1. The contest Kangaroo in Europe has taken place every year since So, the contest Kangaroo in 2006 is the A) 15 th B) 16 th C) 17 th D) 14 th (0+6) (20 0)+6 = A) 106 B) 114 C) 126 D) The point O is the center of a regular pentagon. How much of the pentagon is shaded? A) 20% B) 25% C) 30% D) 40% O 4. Fatima told her grandchildren: If I give 2 toffees each of you I am left with 3 toffees. But if I try to give 3 toffees each of you I face a short of 2 toffees. How many grandchildren does Fatima have? A) 3 B) 4 C) 5 D) 6 5. A cube with two holes in the right picture has one of the following nets: A) B) C) D) 6. An interview of 2006 schoolchildren from Minsk (capital of Belorussia) revealed that 1500 of them participated in the "Kangaroo" contest, in the "Bear cub" competition. How many from the interviewed children participated in both competitions, if 6 of them did not participate in either of the competitions? A) 300 B) 500 C) 600 D) 700

8 Kangaroo 2006 Cadet Max Time: 95 min 7. The solid in the picture is created from two cubes. The small cube with edges 1 cm long is placed on the top of a bigger cube with edges 3 cm long. What is the surface area of this solid? A) 56 cm 2 B) 58 cm 2 C) 59 cm 2 D) 60 cm 2 8. A container that can hold 12 litres is 4 3 full. How much will it contain after 4 litres has been poured out of it? A) 5 B) 6 C) 8 D) Two sides of a triangle are each 7 cm long. The length of the third side is an integer number of centimeters. At most how many centimeters do the perimeter of the triangle measure? A) 14 B) 15 C) 21 D) An air temperature fell 10 o in the night but has risen twice in the day and became the same. Find the night temperature of the air. A) 50 B) 40 C) 30 D) 20 4-Point-Problems 11. If it s blue, it s round. That means: If it s square, it s red. It s either blue or yellow. If it s yellow, it s square. It s either square or round. A) It s red and round B) It s a blue square C) It s blue and round D) It s yellow and round 12. Three Tuesdays of a month fall on even dates. What day of a week was the 21 st day of this month? A) Wednesday B) Thursday C) Friday D) Sunday 13. Ahmad, Babar and Nizami saved money to buy a tent for a camping trip. Nizami saved 60 % of the price. Ahmad saved 40 % of what was left of the price. This way Babar share of the price was 30 Rs. What was the price of the tent? A) 60 Rs B) 125 Rs C) 150 Rs D) 200 Rs 14. Several strange spacemen are traveling through the space in their rocket STAR 1. They are of three colors: green, orange or blue. Green men have two arms, orange men have three arms and blue men have five arms. In the spaceship there are as many green men as orange ones and 10 more blue ones than green ones. Altogether they have 250 arms. How many blue men are traveling in the rocket? A) 15 B) 20 C) 30 D) If kangaroo pushes himself with his left leg, he will jump on 2 m, if he pushes with the right leg, he will jump on 4 m, and if he pushes with both legs, he will jump on 7 m. What the least number of jumps should kangaroo make to cover a distance of exactly 1000 m? A) 142 B) 143 C) 144 D) 250

9 Kangaroo 2006 Cadet Max Time: 95 min 16. Which number when squared is increased by 500%? A) 5 B) 6 C) 7 D) Raza and Ijaz have drawn a 4x4-square and marked the centers of the small squares. Afterwards, they draw obstacles and then find out in how many ways it is possible to go from A to B using the shortest way avoiding the obstacles and going from centre to centre only vertically and horizontally. How many shortest paths are there from A to B under these conditions? A) 6 B) 8 C) 9 D) Amna counts her fingers in such a way (see figure). So the first finger has a set of numbers: 1, 9, 17 Find the first number for a finger with number 2006 A) 1 B) 2 C) 3 D) Find a truly end of the sentence: If your reflection looks on me then A) you look on mine reflection B) my reflection looks on you C) you look on me D) I look on your reflection 20. One fruit of guava has the same number of seeds as 2 tangerines and 11 apples have. One half of guava has the same number of seeds as 4 apples and 3 tangerines. How many apples have the same sum of seeds as 100 tangerines? A) 25 B) 50 C) 75 D) Point-Problems 21. Shaheen is making patterns with toothpicks according to the schema of the figure. How many toothpicks does Shaheen add to the 10th pattern to make the 11st? A) 40 B) 42 C) 44 D) A train is composed of four wagons, I, II, III and IV, pulled by a locomotive. In how many ways can the train be composed so that the wagon I is nearer the locomotive that the wagon II? A) 4 B) 12 C) 24 D) If the sum of three positive numbers is equal to 20, then the product of the two largest numbers among them cannot be A) greater than 99 B) less than C) equal to 25 D) equal to 100

10 Kangaroo 2006 Cadet Max Time: 95 min 24. The natural numbers from 1 to 2006 are written down on the blackboard. Akhlaq underlined all numbers divisible by 2, then all numbers divisible by 3, and then all numbers divisible by 4. How many numbers are underlined precisely twice? A) 1003 B) 1002 C) 501 D) A house has 10 rooms. Ten boys stay in different rooms and count the number of doors in them. After that they sum all results and receive 25. What a proposition can't be true about number N of doors which led outside the house? A) N=7 B) N=5 C) N=3 D) N=2

11 KANGAROO-2006: Correct Answers N Ecolier Benjamin Cadet Junior Student 1 B B B D A 2 A D C C B 3 B B C D E 4 A D C E C 5 A C D E B 6 B C D C D 7 B A B D C 8 A D A E B 9 B D D A D 10 B C D D E 11 B D C B D 12 A B D B B 13 B B B D A 14 A D C A A 15 A C C C E 16 B B B A A 17 B C D E D 18 B D D E B 19 A D A A D 20 B C C B B 21 C E A 22 B B B 23 D B D 24 C A C 25 D C C 26 C E 27 C B 28 C A 29 C C 30 B A

12 INTERNATIONAL KANGAROO MATHEMATICS CONTEST 2007 Level Cadet: Class (7 & 8) Max Time: 1 Hour & 35 Min 3-Point-Problems Q = A) 1003 B) 223 C) 213 D) 123 Q2. Rose plants were planted in a line on both sides of the path. The distance between each plant was 2 m. What is the maximum number of plants that were planted if the path is 20 m long? A) 22 B) 20 C) 12 D) 11 Q3. The robot starts walking on the table from the place A2 in the direction of arrow, as shown on the picture. It can go always forward. If it meets with difficulties (black boxes and the boundary), it turns right. The robot will stop in case, if he can t go forward after turning right. On which place will it stop A) B2 B) A1 C) E1 D) nowhere Q4. What is the sum of the points on the invisible faces of the dice? A) 15 B) 12 C) 7 D) 27 Q 5. If the sum of two positive integers is 11, then the maximum of their product will be A) 24 B) 28 C) 30 D) 32 Q6. A small square is inscribed in a big one as shown in the figure. Find the area of the small square A) 16 B) 28 C) 34 D) 36 1 of 4

13 INTERNATIONAL KANGAROO MATHEMATICS CONTEST 2007 Q7. At least how many little squares we have to shade in the picture on the right so that it has an axis of symmetry? A) 3 B) 5 C) 2 D) 4 Q8. A palindromic number is one that reads the same backwards as forwards, so is a palindromic number. What is the difference between the smallest 5-digit palindromic number and the largest 6-digit palindromic numbers? A) B) C) D) Q9. On the picture, there are six identical circles. The circles touch the sides of a large rectangle and each other as well. The vertices of the small rectangle lie in the centres of the four circles. The circumference of the small rectangle is 60 cm. What is the circumference of the large rectangle? A) 160 cm B) 120 cm C) 100 cm D) 80 cm Q10. x is a strictly negative integer. Which is the biggest? A) -2x B) 2x C) 6x+2 D) x 2 4-Point-Problems Q11. The squares are formed by intersecting the segment AB of length 24 cm by the broken line AA 1 A 2... A 12 B (see the Fig.). Find the length of AA 1 A 2... A 12 B. A) 48 cm B) 72 cm C) 96 cm D) 106 cm 2 of 4

14 INTERNATIONAL KANGAROO MATHEMATICS CONTEST 2007 Q12. On parallel lines l 1 and l 2, 6 points were drawn; 4 on line l 1 and 2 on line l 2. What is the total number of triangles whose vertices are given points? A) 6 B) 12 C) 16 D) 18 Q13. A survey found that 2/3 of all customers buy product A and 1/3 buy product B. After a publicity campaign for product B a new survey showed that 1/4 of the customers who preferred product A are now buying product B. So now we have A) 1/4 of the customers buy product A, 3/4 buy product B B) 7/12 of the customers buy product A, 5/12 buy product B C) 1/2 of the customers buy product A, 1/2 buy product B D) 1/3 of the customers buy product A, 2/3 buy product B Q14. In order to obtain the number 8 8, we must raise 4 4 to the power A) 3 B) 2 C) 4 D) 8 Q15. ABC and CDE are equal equilateral triangles. If angle ACD = 80 o, what is angle ABD? A) 25 o B) 30 o C) 35 o D) 40 o Q16. Look at the numbers 1, 2, 3, 4,..., 100. How many percent of these numbers is a perfect square? A) 1% B) 5% C) 25% D) 10% Q17. By drawing 9 line segments (5 horizontal and 4 vertical) as shown in figure, Amir has made a table of 12 cells. If he had used 6 horizontal and 3 vertical lines, he would have got 10 cells only. How many cells you can get maximally if you draw at most 15 lines? A) 30 B) 36 C) 40 D) 42 Q18. How many possible routes with the minimum number of moves are there for a man to travel from A to B of the grid (man can move to any adjacent square, including diagonally) A B A) 4 B) 3 C) 5 D) 2 3 of 4

15 INTERNATIONAL KANGAROO MATHEMATICS CONTEST 2007 Q19. If you choose three numbers from the grid shown, so that you have one number from each row and also have one number from each column, and then add the three numbers together, what is the largest total that can be obtained? A) 18 B) 15 C) 21 D) 24 Q20. The segments OA, OB, OC and OD are drawn from the center O of the square KLMN to its sides so that AOB=90 o and COD=90 o (as shown in the figure). If the side of the square equals 2, the area of the shaded part equals. A) 1 B) 2 C) 2.5 D) Point-Problems Q21. A broken calculator does not display the digit 1. For example, if we type in the number 3131, only the number 33 is displayed, with no spaces. Awais typed a 6-digit number into that calculator, but only 2007 appeared on the display. How many numbers could have Awais typed? A) 12 B) 13 C) 14 D) 15 Q22. The first digit of a 4-digit number is equal to the number of zeros in this number, the second digit is equal to the number of digits 1, the third digit is equal to the number of digits 2, the fourth - the number of digits 3. How many such numbers exist? A) 3 B) 2 C) 4 D) 5 Q23. A positive integer number n has 2 divisors, while n+1 has 3 divisors. How many divisors does n + 2 have? A) 2 B) 3 C) 4 D) 5 Q24. The table 3 3 contains natural numbers (see picture). Nasir and Ali crossed out four numbers each so that the sum of the numbers crossed out by Nasir is three times as great as the sum of the numbers, crossed out by Ali. The number which remained in the table after crossing is: A) 4 B) 14 C) 23 D) 24 Q25. Five integers are written around a circle in such a way that no two or three consecutive numbers give a sum divisible by 3. Among those 5 numbers, how many are divisible by 3? A) 0 B) 1 C) 2 D) 3 4 of 4

16 KANGAROO-2007: Correct Answers N Eclolier Benjamin Cadet Junior Student 1 B C B E A 2 B B A C A 3 B A D B A 4 A C D D E 5 A D C B C 6 A B C C C 7 A A A B B 8 B D B D A 9 A C C B E 10 A D A B B 11 A B B C C 12 B B C B D 13 A C C B B 14 B D A C E 15 B A D A C 16 B A D B D 17 A D D E C 18 A B A D E 19 A C B D A 20 B A B A D 21 D C D 22 B D D 23 A A D 24 B D B 25 C B B 26 D C 27 C C 28 C C 29 D A 30 B B

17 International Kangaroo Mathematics Contest 2008 Cadet Level: Class (7 & 8) Max Time: 2 Hours 3-point problems 1) How many pieces of string are there in the picture? 2) A) 3 B) 4 C) 5 D) 6 In a class there are 9 boys and 13 girls. Half of the children in this class have got a cold. How many girls at least have a cold? 3) A) 0 B) 1 C) 2 D) 3 6 kangaroos eat 6 sacks of grass in 6 minutes. How many kangaroos will eat 100 sacks of grass in 100 minutes? A) 100 B) 60 C) 6 D) 600 4) Numbers 2, 3, 4 and one more number are written in the cells of 2 2 table. It is known that the sums of the numbers in the first row are equal to 9, and the sum of the numbers in the second row is equal to 6. The unknown number is A) 5 B) 6 C) 7 D) 8 5) The triangle and the square have the same perimeter. What is the perimeter of the whole figure (a pentagon)? 4cm A) 24 cm B) 28 cm C) 32 cm D) It depends upon the triangle measures

18 6) A florist had 24 white, 42 red and 36 yellow roses left. At most, how many identical bunches can she make, if she wants to use all the remaining flowers? A) 4 B) 6 C) 8 D) 12 7) A cube has all its corners cut off, as shown. How many edges does the resulting shape have? A) 30 B) 36 C) 40 D) Another answer 8) Three lines intersect in one point. Two angles are given in the figure. How many degrees is the grey angle? A) 52 B) 53 C) 54 D) ) Ali has 9 coins (each is worth 2 cents); while his sister Saima has 8 coins, each being 5 cents. What the least number of coins they should interchange (with each other) in order to equalize their money? A) 4 B) 5 C) 12 D) it is impossible to do 10) How many squares can be drawn by joining the dots with line segments? A) 2 B) 3 C) 4 D) 5 4-point problems 11) If there are two buses on the circular bus route, the interval between them is 25 min. How many extra buses are necessary to shorten the interval by 60%? A) 2 B) 3 C) 5 D) 6 12) The French mathematician August de Morgan claimed that he was х years old in the year of х 2. He is known to have died in When was he born? A) 1806 B) 1848 C) 1849 D) another answer

19 13) We decide to visit by ferry-boat four islands A,B,C & D starting from the mainland. B can be reached only from A or from the mainland, A & C are connected to each other and with the mainland and D is connected only with A. Which is the minimum number of ferry runs that we need, if we want to visit all the islands? A) 6 B) 5 C) 4 D) 7 14) Tom and Jerry cut two equal rectangles. Tom got two rectangles with the perimeter of 40 cm each, and Jerry got two rectangles with the perimeter of 50 cm each. What were the perimeters of the initial rectangles? A) 40 cm B) 50 cm C) 60 cm D) 80 cm 15) One of the cube faces is cut along its diagonals (see the fig.). Which of the following net is impossible? A) 1 and 3 B) 1 and 5 C) 3 and 4 D) 3 and 5 16) Points A, B, C and D are marked on the straight line in some order. It is known that AB = 13, BC = 11, CD = 14 and DA = 12. What is the distance between the farthest two points? A) 14 B) 38 C) 25 D) another answer 17) Four tangent congruent circles of radius 6 cm are inscribed in a rectangle. If P is a vertex and Q and R are points of tangency, what is the area of triangle PQR? Q P A) 27 cm 2 B) 45 cm 2 C) 54 cm 2 D) 108 cm 2 R 18) Seven cards lie in a box. Numbers from 1 to 7 are written on these cards (exactly one number on the card). The first sage takes, at random, 3 cards from the box and the second sage takes 2 cards (2 cards are left in the box). Then the first sage tells to the second one: I know that the sum of the numbers of your cards is even. The sum of card s numbers of the first sage is equal to A) 10 B) 12 C) 9 D) 15

20 19) In an isosceles triangle ABC, the bisector CD of the angle C is equal to the base BC. Then the angle CDA is equal to A) 100º B) 108º C) 120º D) impossible to determine 20) A wooden cube 11 x 11 x 11 is obtained by sticking together 11 3 unit cubes. What is the largest number of unit cubes visible from a same point of view? A) 329 B) 330 C) 331 D) point problems 21) In the equality KAN GAR = OO any letter stands for some digit (different letters for different digits, equal letters for equal digits). Find the largest possible value of the number KAN? A) 876 B) 865 C) 864 D) ) A boy always speaks the truth on Thursday and Fridays, always tells lies on Tuesdays, and randomly tells the truth or lies on other days of the week. On seven consecutive days he was asked what his name was, and on the first six days he gave the following answers in order: Akbar, Ali, Akbar, Ali, Farooq, Ali. What did he answer on the seventh day? A) Akbar B) Ali C) Amir D) another answer 23) Four identical dice are arranged in a row (see the fig.). The dice are not standard, i.e., the sum of points in the opposite faces of the dice not necessarily equals 7. Find the total sum of the points in all 6 touching faces of the dice. A) 19 B) 20 C) 21 D) 22 24) Some straight lines are drawn on the plane so that all angles 10, 20, 30, 40, 50, 60, 70, 80, 90 are among the angles between these lines. Determine the smallest possible number of these straight lines. A) 4 B) 5 C) 6 D) 7 25) On my first spelling test, I score one mark out of five. If I now work hard and get full marks on every test, who many more tests should I take for my average to be four out of five correct answers? A) 2 B) 3 C) 4 D) 5 GOOD LUCK!

21 Inernational Mathematics Contest 2008 Answer of Questions Q. No. Ecolier Level Q. No. Benjamin Level Q. No. Cadet Level Q. No. Junior Level Q. No. Student Level 1 B 1 C 1 A 1 D 1 B 2 B 2 C 2 C 2 C 2 C 3 A 3 B 3 C 3 B 3 B 4 A 4 C 4 B 4 B 4 B 5 A 5 B 5 A 5 B 5 D 6 B 6 D 6 B 6 D 6 D 7 A 7 A 7 B 7 A 7 E 8 A 8 D 8 A 8 C 8 C 9 B 9 D 9 B 9 C 9 C 10 B 10 C 10 B 10 B 10 E 11 B 11 B 11 B 11 B 11 E 12 A 12 A 12 A 12 D 12 A 13 A 13 D 13 B 13 D 13 D 14 A 14 A 14 C 14 B 14 B 15 A 15 C 15 D 15 A 15 A 16 A 16 D 16 C 16 A 16 A 17 A 17 D 17 D 17 B 17 C 18 B 18 B 18 B 18 C 18 A 19 B 19 D 19 B 19 A 19 B 20 B 20 C 20 C 20 B 20 B 21 C 21 D 21 C 22 A 22 E 22 A 23 B 23 D 23 B 24 B 24 D 24 C 25 B 25 B 25 E 26 B 26 D 27 C 27 B 28 A 28 E 29 E 29 B 30 D 30 A Answer of IKMC 2008

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27 International Kangaroo Mathematics Contest 2009 Answer of Questions Q. No. Ecolier Level Q. No. Benjamin Level Q. No. Cadet Level Q. No. Junior Level Q. No. Student Level 1 B 1 D 1 D 1 C 1 E 2 B 2 B 2 C 2 C 2 A 3 A 3 C 3 C 3 A 3 B 4 B 4 A 4 B 4 C 4 C 5 B 5 D 5 C 5 D 5 C 6 A 6 C 6 D 6 C 6 D 7 B 7 C 7 C 7 B 7 D 8 A 8 D 8 C 8 B 8 E 9 A 9 C 9 B 9 E 9 D 10 A 10 C 10 C 10 B 10 D 11 A 11 B 11 B 11 C 11 B 12 B 12 D 12 C 12 B 12 B 13 B 13 D 13 C 13 C 13 B 14 B 14 C 14 A 14 C 14 C 15 B 15 C 15 B 15 C 15 E 16 B 16 D 16 C 16 C 16 B 17 A 17 B 17 A 17 D 17 D 18 B 18 D 18 D 18 A 18 B 19 B 19 C 19 D 19 C 19 D 20 A 20 A 20 C 20 C 20 D 21 B 21 C 21 D 22 C 22 B 22 C 23 A 23 C 23 C 24 C 24 C 24 B 25 B 25 B 25 B 26 C 26 A 27 A 27 B 28 B 28 C 29 C 29 B 30 D 30 B Answer of IKMC 2009.xls

28 International Kangaroo Mathematics Contest 2010 Cadet Level: Class (7 & 8) Max Time: 2 Hours 3-point problems Q1) How much is ? A) 396 B) 404 C) 405 D) other answer. Q2) How many axes of symmetry does the figure have? A) 1 B) 2 C) 4 D) infinitely many Q3) Toy kangaroos are packed for shipment. Each of them is packed in a box which is a cube. Exactly eight boxes are packed tightly in a bigger cubic cardboard box. How many kangaroo boxes are on the bottom floor of this big cube? A) 1 B) 2 C) 3 D) 4 Q4) The perimeter of the figure is equal to A) 3a + 4b B) 3a + 8b C) 6a + 6b D) 6a + 8b Q5) Ehsan draws the six vertices of a regular hexagon and then connects some of the 6 points with lines to obtain a geometric figure. Then this figure is surely not a A) trapezium B) right angled triangle C) square D) obtuse angled triangle Q6) If we type seven consecutive integer numbers and the sum of the smallest three numbers is 33, which is the sum of the largest three numbers? A) 39 B) 42 C) 48 D) 45 Q7) After stocking up firewood, the worker summed up that from the certain number of logs he made 72 logs besides 53 cuts were made. He saws only one log at a time. How many logs were at the beginning? A) 18 B) 19 C) 20 D) 21 1

29 Q8) There are seven bars in the box. They are 3 cm 1 cm in size. The box is of size 5 cm 5 cm. Is it possible to slide the bars in the box so there will be room for one more bar? At least how many bars must be moved in that case? A) 2 B) 3 C) 4 D) It is impossible Q9) A square is divided into 4 smaller equal-sized squares. All the smaller squares are coloured either green or blue. How many different ways are there to colour the given square? (Two colourings are considered the same if one can be rotated to give the other.) A) 5 B) 6 C) 7 D) 8 Q10) The sum of the first hundred positive odd integers subtracted from sum of the first hundred positive even integers is A) 0 B) 50 C) 100 D) point problems Q11) Grandma baked a cake for her grandchildren who will visit in the afternoon. Unfortunately she forgot whether only 3, 5 or all 6 of her grandchildren will come over. She wants to ensure that every child gets the same amount of cake. Then, to be prepared for all three possibilities she better cut the cake into A) 12 pieces B) 15 pieces C) 18 pieces D) 30 pieces Q12) Which of the following is the smallest two-digit number that is not the sum of three different onedigit numbers? A) 10 B) 15 C) 23 D) 25 Q13) Fatima needs 18 min to make a long chain by connecting three short chains with extra chain links. How long does it take her to make a really long chain by connecting six short chains in the same way? A) 27 min B) 30 min C) 36 min D) 45 min Q14) In quadrilateral ABCD we have AD = BC, DAC = 50º, DCA = 65º, ACB = 70º (see the fig.). Find the value of ABC. A) 55º B) 60º C) 65º D) impossible to determine. 2

30 Q15) Saima has wound some rope around a piece of wood. She rotates the wood as shown with the arrow. What does she see after the rotation? Front side A) B) C) D) Q16) There are 50 bricks of white, blue and red colour in the box. The number of white bricks is eleven times the number of blue ones. There are fewer red ones than white ones, but more red ones than blue ones. How many fewer red bricks are there than white ones? A) 2 B) 11 C) 19 D) 22 Q17) On the picture ABCD is a rectangle, PQRS is a square. The shaded area is half of the area of rectangle ABCD. What is the length of PX? A) 1 B) 1.5 C) 2 D) 4 Q18) What is the smallest number of straight lines needed to divide the plane into exactly 5 regions? A) 3 B) 4 C) 5 D) another answer Q19) If a 1 = b + 2 = c 3 = d + 4 = e 5, then which of the numbers a, b, c, d, e is the largest? A) a B) c C) d D) e Q20) The logo shown is made entirely from semicircular arcs of radius 2 cm, 4 cm or 8 cm. What fraction of the logo is shaded? A) 1 3 B) 1 4 C) 1 5 D) 3 4 3

31 5-point problems Q21) In the figure there are nine regions inside the circles. Put all the numbers from 1 to 9 exactly one in each region so that the sum of the numbers inside each circle is 11. Which number must be written in the region with the question mark? A) 5 B) 6 C) 7 D) 8 Q22) A paper strip was folded three times in half and then completely unfolded so that you can still see the 7 folds going up or down. Which of the following views from the side cannot be obtained in this way? Q23) On each of 18 cards exactly one number is written, either 4 or 5. The sum of all numbers on the cards is divisible by 17. On how many cards is the number 4 written? A) 4 B) 5 C) 6 D) 7 Q24) The natural numbers from 1 to 10 are written on the blackboard. The students in the class play the following game: a student deletes 2 of the numbers and instead of them writes on the blackboard their sum decreased by 1; after that another student deletes 2 of the numbers and instead of them writes on the blackboard their sum decreased by 1; and so on. The game continues until only one number remains on the blackboard. The last number is: A) less than 11 B) 11 C) 46 D) greater than 46 Q25) A Kangaroo has a large collection of small cubes Each cube is a single colour. Kangaroo wants to use 27 small cubes to make a cube so that any two cubes with at least one common vertex are of different colours. At least how many colours have to be used? A) 6 B) 8 C) 9 D) 12 4

32 International Kangaroo Mathematics Contest 2010 Answer of Questions Q. No. Ecolier Level Q. No. Benjamin Level Q. No. Cadet Level Q. No. Junior Level Q. No. Student Level 1 B 1 B 1 B 1 D 1 B 2 A 2 C 2 B 2 D 2 C 3 B 3 C 3 D 3 C 3 D 4 A 4 C 4 D 4 B 4 D 5 A 5 C 5 C 5 D 5 D 6 B 6 B 6 D 6 D 6 B 7 B 7 B 7 B 7 E 7 D 8 A 8 B 8 B 8 B 8 E 9 A 9 D 9 B 9 D 9 A 10 A 10 D 10 C 10 A 10 A 11 A 11 C 11 D 11 C 11 E 12 B 12 D 12 D 12 D 12 C 13 B 13 D 13 D 13 B 13 C 14 B 14 B 14 A 14 B 14 A 15 B 15 B 15 B 15 E 15 A 16 B 16 A 16 C 16 C 16 A 17 A 17 A 17 A 17 D 17 B 18 A 18 A 18 B 18 B 18 E 19 A 19 D 19 D 19 C 19 A 20 A 20 B 20 B 20 A 20 C 21 B 21 B 21 B 22 D 22 C 22 D 23 B 23 C 23 D 24 C 24 B 24 E 25 B 25 D 25 B 26 A 26 B 27 C 27 C 28 E 28 E 29 A 29 A 30 E 30 C

33 3 point problems PROBLEM 01 Which of the following has the largest value? (A) 2011 (B) 1 (C) (D) (E) PROBLEM 02 Elsa plays with tetrahedrons and cubes. She has 5 cubes and 3 tetrahedrons. How many faces are there in total? (A) 42 (B) 48 (C) 50 (D) 52 (E) 56 PROBLEM 03 A zebra crossing has alternate white and black stripes, each of width 50 cm. The crossing starts and ends with a white stripe and has 8 white stripes in all. What is the total width of the crossing? (A) 7 m (B) 7.5 m (C) 8 m (D) 8.5 m (E) 9 m PROBLEM 04 My broken calculator divides instead of multiplying and subtracts instead of adding. I type (12 3) + (4 2). What answer does the calculator show? (A) 2 (B) 6 (C) 12 (D) 28 (E) 38 PROBLEM 05 My digital watch has just changed to show the time 20:11. How many minutes later will it next show a time with the digits 0, 1, 1, 2 in some order? (A) 40 (B) 45 (C) 50 (D) 55 (E) 60 PROBLEM 06 The diagram shows three squares. The medium square is formed by joining the midpoints of the large square. The small square is formed by joining the midpoints of the medium square. The area of the small square in the figure is 6 cm 2. What is the difference between the area of the medium square and the area of the large square? (A) 6 cm 2 (B) 9 cm 2 (C) 12 cm 2 (D) 15 cm 2 (E) 18 cm 2 Level Cadet Class 7 & 8

34 PROBLEM 07 In my street there are 17 houses. On the `even' side, the houses are numbered 2, 4, 6, and so on. On the `odd' side, the houses are numbered 1, 3, 5, and so on. I live in the last house on the even side, which is number 12. My cousin lives in the last house on the odd side. What is the number of my cousin's house? (A) 5 (B) 7 (C) 13 (D) 17 (E) 21 PROBLEM 08 Felix the Cat caught 12 fish in three days. Each day after the first he caught more fish than the previous day. On the third day he caught fewer fish than the first two days together. How many fish did Felix catch on the third day? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 PROBLEM 09 Mary lists every 3-digit number whose digits add up to 8. What is the sum of the largest and the smallest numbers in Mary's list? (A) 707 (B) 907 (C) 916 (D) 1000 (E) 1001 PROBLEM 10 The diagram shows an L-shape made from four small squares. Ria wants to add an extra small square in order to form a shape with a line of symmetry. In how many different ways can she do this? (A) 1 (B) 2 (C) 3 (D) 5 (E) 6 4 point problems PROBLEM 11 What is the value of...? (A) 0.01 (B) 0.1 (C) 1 (D) 10 (E) 100 PROBLEM 12 Marie has 9 pearls that weigh 1 g, 2 g, 3 g, 4 g, 5 g, 6 g, 7 g, 8 g, and 9 g. She makes four rings, using two pearls on each. The total weight of the pearls on each of these four rings is 17 g, 13 g, 7 g and 5 g, respectively. What is the weight of the unused pearl? (A) 1 g (B) 2 g (C) 3 g (D) 4 g (E) 5 g Level Cadet Class 7 & 8

35 PROBLEM 13 Hamster Fridolin sets out for the Land of Milk and Honey. His way to the legendary Land passes through a system of tunnels. There are 16 pumpkin seeds spread through the tunnels, as shown in the picture. What is the highest number of pumpkin seeds Fridolin can collect if he is not allowed to visit any junction more than once? (A) 12 (B) 13 (C) 14 (D) 15 (E) 16 PROBLEM 14 Each region in the figure is coloured with one of four colours: red (R), green (G), orange (O), or yellow (Y). (The colours of only three regions are shown.) Any two regions that touch have different colours. The colour of the region X is (A) red (B) orange (C) green (D) yellow (E) impossible to determine PROBLEM 15 A teacher has a list of marks: 17, 13, 5, 10, 14, 9, 12, 16. Which two marks can be removed without changing the average? (A) 12 and 17 (B) 5 and 17 (C) 9 and 16 (D) 10 and 12 (E) 10 and 14 PROBLEM 16 A square piece of paper is cut into six rectangular pieces. When the perimeter lengths of the six pieces are added together the result is 120 cm. What is the area of the square piece of paper? (A)48 cm 2 (B) 64 cm 2 (C) cm 2 (D) 144 cm 2 (E) 256 cm 2 Level Cadet Class 7 & 8

36 PROBLEM 17 In three games FC Barcelona scored three goals and let one goal in. In these three games, the club won one game, drew one game and lost one game. What was the score in the game FC Barcelona won? (A) 2-0 (B) 3-0 (C) 1-0 (D) 2-1 (E) 0-1 PROBLEM 18 Lali draws a line segment DE of length 2 cm on a piece of paper. How many different points F can she draw on the paper so that the triangle DEF is right-angled and has area 1 cm 2? (A) 2 (B) 4 (C) 6 (D) 8 (E) 10 PROBLEM 19 The positive number a is less than 1, and the number b is greater then 1. Which of the following numbers has the largest value? (A) a b (B) a + b (C) a b (D) b (E) the answer depends on a and b. PROBLEM 20 The figure shows a net which is cut out of paper and folded to make a cube. A dark line is then drawn on the cube, as shown, dividing the surface of the cube into two identical parts. The cube is then unfolded. The paper could now look like only one of the following. Which one? (A) (B) (C) (D) (E) 5 point problems PROBLEM 21 The five-digit number `24X8Y' is divisible by 4, 5 and 9. What is the sum of the digits X and Y? (A) 13 (B) 10 (C) 9 (D) 5 (E) 4 Level Cadet Class 7 & 8

37 PROBLEM 22 Lina has fixed two shapes on a 5 5 board, as shown in the picture. Which of the following 5 shapes should she place on the empty part of the board so that none of the remaining 4 shapes will fit in the empty space that is left? (The shapes may be rotated or turned over, but can only be placed so that they cover complete squares.) (A) (B) (D) (E) (C) PROBLEM 23 The three blackbirds Isaac, Max and Oscar are each on their own nest. Isaac says: ``I am more than twice as far away from Max as I am from Oscar. Max says: I am more than twice as far away from Oscar as I am from Isaac. Oscar says: I am more than twice as far away from Max as I am from Isaac. At least two of them are telling the truth. Who is lying? (A) Isaac (B) Max (C) Oscar (D) none of them (E) impossible to tell PROBLEM 24 The figure shows a square with side 3 cm inside a square with side 7 cm, and another square with side 5 cm which intersects the first two squares. What is the difference between the area of the black region and the total area of the grey regions? (A) 0 cm 2 (B) 10 cm 2 (C) 11 cm 2 (D) 15 cm 2 (E) impossible to determine PROBLEM 25 Myshko shot at a target. When he hit the target he only hit 5, 8 and 10. Myshko hit 8 and 10 the same number of times. He scored 99 points in total, and 25\% of his shots missed the target. How many times did Myshko shoot at the target? (A) 10 (B) 12 (C) 16 (D) 20 (E) 24 PROBLEM 26 In a convex quadrilateral ABCD with AB = AC, the following angles are known: BAD = 80, ABC = 75, ADC = 65. What is the size of BDC? (A) 10 (B) 15 (C) 20 (D) 30 (E) 45 Level Cadet Class 7 & 8

38 PROBLEM 27 Seven years ago Evie's age was a multiple of 8, and in eight years' time her age will be a multiple of 7. Eight years ago Raph's age was a multiple of 7, and in seven years' time his age will be a multiple of 8. Neither Evie nor Raph is over a hundred years old. Which of the following statements is true? (A) Raph is two years older than Evie (B) Raph is one year older than Evie (C) Raph and Evie are the same age (D) Raph is one year younger than Evie (E) Raph is two years younger than Evie PROBLEM 28 In the expression different letters stand for different non-zero digits, but the same letter always stands for the same digit. What is the smallest possible positive integer value of the expression? (A) 1 (B) 2 (C) 3 (D) 5 (E) 7 PROBLEM 29 The left-hand figure shows a shape consisting of two rectangles. The lengths of two sides are marked: 11 and 13. The shape is cut into three parts and the parts are rearranged into a triangle, as shown in the right-hand figure. What is the length marked x? (A) 36 (B) 37 (C) 38 (D) 39 (E) 40 PROBLEM 30 Mark plays a computer game on a 4 4 grid. Initially the 16 cells are all white; clicking one of the white cells changes it to either red or blue. Exactly two cells will become blue and they will always have a side in common. The aim is to make both blue cells appear in as few clicks as possible. With perfect play, what is the largest number of clicks Mark will ever need to make? (A) 9 (B) 10 (C) 11 (D) 12 (E) 13 Level Cadet Class 7 & 8

39 International Kangaroo Mathematics Contest 2011 Answer of Questions Q. No. Ecolier Level Q. No. Benjamin Level Q. No. Cadet Level Q. No. Junior Level Q. No. Student Level 1 C 1 C 1 D 1 B 1 A 2 C 2 C 2 A 2 C 2 B 3 B 3 A 3 B 3 D 3 A 4 E 4 B 4 A 4 A 4 C 5 B 5 E 5 C 5 E 5 D 6 A 6 E 6 C 6 C 6 C 7 D 7 D 7 E 7 D 7 C 8 C 8 A 8 A 8 C 8 C 9 B 9 E 9 B 9 C 9 C 10 B 10 B 10 C 10 E 10 B 11 B 11 D 11 C 11 B 11 D 12 E 12 B 12 C 12 B 12 E 13 D 13 E 13 B 13 A 13 D 14 A 14 C 14 A 14 B 14 D 15 C 15 D 15 E 15 C 15 B 16 B 16 B 16 D 16 D 16 A 17 C 17 B 17 B 17 D 17 E 18 C 18 C 18 C 18 D 18 D 19 C 19 D 19 B 19 C 19 A 20 E 20 E 20 A 20 B 20 B 21 D 21 C 21 E 21 C 21 B 22 A 22 C 22 D 22 C 22 C 23 E 23 D 23 B 23 B 23 A 24 D 24 C 24 D 24 C 24 A 25 D 25 D 25 B 25 D 26 E 26 D 26 D 26 B 27 D 27 A 27 B 27 C 28 A 28 B 28 C 28 D 29 D 29 B 29 C 29 C 30 A 30 B 30 E 30 C

40 International Kangaroo Mathematics Contest 2012 Cadet Level Cadet (Class 7 & 8) Time Allowed : 3 hours SECTION ONE - (3 points problems) 1. Four chocolate bars cost 6 EUR more than one chocolate bar. What is the cost of one chocolate bar? (A) 1 EUR (B) 2 EUR (C) 3 EUR (D) 4 EUR (E) 5 EUR = (A) (B) (C) 9.99 (D) (E) A watch is placed face up on a table so that its minute hand points north-east. How many minutes pass before the minute hand points north-west for the first time? (A) 45 (B) 40 (C) 30 (D) 20 (E) Mary has a pair of scissors and five cardboard letters. She cuts each letter exactly once (along a straight line) so that it falls apart in as many pieces as possible. Which letter falls apart into the most pieces? (A) (B) (C) (D) (E) 5. A dragon has five heads. Every time a head is chopped off, five new heads grow. If six heads are chopped off one by one, how many heads will the dragon finally have? (A) 25 (B) 28 (C) 29 (D) 30 (E) In which of the following expressions can we replace each occurrence of the number 8 by the same positive number (other than 8) and obtain the same result? (A) (8 + 8) : (B) 8 (8 + 8) : 8 (C) (D) ( ) 8 (E) ( ) : 8 7. Each of the nine paths in a park is 100 m long. Ann wants to go from A to B without going along any path more than once. What is the length of the longest route she can choose? 1 of 7

41 International Kangaroo Mathematics Contest 2012 Cadet B A (A) 900 m (B) 800 m (C) 700 m (D) 600 m (E) 400 m 8. The diagram shows two triangles. In how many ways can you choose two vertices, one in each triangle, so that the straight line through the vertices does not cross either triangle? (A) 1 (B) 2 (C) 3 (D) 4 (E) more than 4 9. Werner folds a sheet of paper as shown in the figure and makes two straight cuts with a pair of scissors. He then opens up the paper again. Which of the following shapes cannot be the result? (A) (B) (C) (D) (E) 10. A cuboid is made of four pieces, as shown. Each piece consists of four cubes and is a single colour. What is the shape of the white piece? (A) (B) (D) (E) (C) 2 of 7

42 International Kangaroo Mathematics Contest 2012 Cadet SECTION TWO - (4 points problems) 11. Kanga forms two 4-digit natural numbers using each of the digits 1, 2, 3, 4, 5, 6, 7 and 8 exactly once. Kanga wants the sum of the two numbers to be as small as possible. What is the value of this smallest possible sum? (A) 2468 (B) 3333 (C) 3825 (D) 4734 (E) Mrs Gardner grows peas and strawberries. This year she has changed the rectangular pea bed to a square by lengthening one of its sides by 3 metres. As a result of this change, the area of the strawberry bed was reduced by 15 m 2. What was the area of the pea bed before Last year This year Peas Peas the change? Strawberries Strawberries (A) 5 m 2 (B) 9 m 2 (C) 10 m 2 (D) 15 m 2 (E) 18 m Barbara wants to complete the diagram by inserting three numbers, one in each empty cell. She wants the sum of the first three numbers to be 100, the sum of the three middle numbers to be 200 and the sum of the last three numbers to be 300. What number should Barbara insert in the middle cell of the diagram? (A) 50 (B) 60 (C) 70 (D) 75 (E) In the figure, what is the value of x? 3 of 7

43 International Kangaroo Mathematics Contest 2012 Cadet x (A) 35 (B) 42 (C) 51 (D) 65 (E) Four cards each have a number written on one side and a phrase written on the other. The four phrases are divisible by 7, prime, odd and greater than 100, and the four numbers are 2, 5, 7 and 12. On each card, the number does not correspond to the phrase on the other side. What number is written on the same card as the phrase greater than 100? (A) 2 (B) 5 (C) 7 (D) 12 (E) impossible to determine 16. Three small equilateral triangles of the same size are cut from the corners of a larger equilateral triangle with sides of 6 cm, as shown. The sum of the perimeters of the three small triangles is equal to the perimeter of the remaining grey hexagon. What is the side length of the small triangles? (A) 1 cm (B) 1.2 cm (C) 1.25 cm (D) 1.5 cm (E) 2 cm 17. A piece of cheese is cut into a large number of pieces. During the course of the day, a number of mice came and stole some pieces, watched by the lazy cat Ginger. Ginger noticed that each mouse stole a different number of pieces less than 10, and that no mouse stole exactly twice as many pieces as any other mouse. What is the largest number of mice that Ginger 4 of 7

44 International Kangaroo Mathematics Contest 2012 Cadet could have seen stealing cheese? (A) 4 (B) 5 (C) 6 (D) 7 (E) At the airport there is a moving walkway 500 metres long, which moves with a speed of 4 km/hour. Ann and Bill step on the walkway at the same time. Ann walks with a speed of 6 km/hour on the walkway while Bill stands still. When Ann comes to the end of the walkway, how far is she ahead of Bill? (A) 100 m (B) 160 m (C) 200 m (D) 250 m (E) 300 m 19. A magical talking square originally has sides of length 8 cm. If he tells the truth, then his sides become 2 cm shorter. If he lies, then his perimeter doubles. He makes four statements, two true and two false, in some order. What is the largest possible perimeter of the square after the four statements? (A) 28 (B) 80 (C) 88 (D) 112 (E) A cube is rolled on a plane so that it turns around its edges. Its bottom face passes through the positions 1, 2, 3, 4, 5, 6, and 7 in that order, as shown. Which two of these positions were occupied by the same face of the cube? (A) 1 and 7 (B) 1 and 6 (C) 1 and 5 (D) 2 and 7 (E) 2 and 6 SECTION THREE - (5 points problems) 21. Rick has five cubes. When he arranges them from smallest to largest, the difference between the heights of any two neighbouring cubes is 2 cm. The largest cube is as high as a tower built from the two smallest cubes. How high is a tower built from all five cubes? (A) 6 cm (B) 14 cm (C) 22 cm (D) 44 cm (E) 50 cm 22. In the diagram ABCD is a square, M is the midpoint of AD and MN is perpendicular D C M N to AC. A B What is the ratio of the area of the shaded triangle MNC 5 of 7

45 International Kangaroo Mathematics Contest 2012 Cadet to the area of the square? (A) 1:6 (B) 1:5 (C) 7:36 (D) 3:16 (E) 7: The tango is danced in pairs, each consisting of one man and one woman. At a dance evening no more than 50 people are present. At one moment 3/4 of the men are dancing with 4/5 of the women. How many people are dancing at that moment? (A) 20 (B) 24 (C) 30 (D) 32 (E) David wants to arrange the twelve numbers from 1 to 12 in a circle so that any two neighbouring numbers differ by either 2 or 3. Which of the following pairs of numbers have to be neighbours? (A) 5 and 8 (B) 3 and 5 (C) 7 and 9 (D) 6 and 8 (E) 4 and Some three-digit integers have the following property: if you remove the first digit of the number, you get a perfect square; if instead you remove the last digit of the number, you also get a perfect square. What is the sum of all the three-digit integers with this curious property? (A) 1013 (B) 1177 (C) 1465 (D) 1993 (E) A book contains 30 stories, each starting on a new page. The lengths of the stories are 1, 2, 3,..., 30 pages. The first story starts on the first page. What is the largest number of stories that can start on an odd-numbered page? (A) 15 (B) 18 (C) 20 (D) 21 (E) An equilateral triangle starts in a given position and is moved to new positions in a sequence of steps. At each step it is rotated about its centre, first by 3, then by a further 9, then by a further 27, and so on (at the n-th step it is rotated by a further (3 n ) ). How many different positions, including the initial position, will the triangle occupy? Two positions are considered equal if the triangle covers the same part of the plane. (A) 3 (B) 4 (C) 5 (D) 6 (E) A rope is folded in half, then in half again, and then in half again. Finally the folded rope is cut through, forming several strands. The lengths of two of the strands are 4 m and 9 m. Which of the following could not have been the length of the whole rope? (A) 52 m (B) 68 m (C) 72 m (D) 88 m (E) all the previous are possible 6 of 7

46 International Kangaroo Mathematics Contest 2012 Cadet 29. A triangle is divided into four triangles and three quadrilaterals by three straight line segments. The sum of the perimeters of the three quadrilaterals is equal to 25 cm. The sum of the perimeters of the four triangles is equal to 20 cm. The perimeter of the whole triangle is equal to 19 cm. What is the sum of the lengths of the three straight line segments? (A) 11 (B) 12 (C) 13 (D) 15 (E) A positive number is to be placed in each cell of the 3 3 grid shown, so that: in each row and each column, the product of the three numbers is equal to 1; and in each 2 2 square, the product of the four numbers is equal to 2. the central cell? What number should be placed in (A) 16 (B) 8 (C) 4 (D) 1 4 (E) of 7

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48 International Kangaroo Mathematics Contest 2012 Answer of Questions Q. No. Pre-Ecolier Level Q. No. Ecolier Level Q. No. Benjamin Level Q. No. Cadet Level Q. No. Junior Level Q. No. Student Level 1 C 1 B 1 C 1 B 1 D 1 E 2 B 2 D 2 C 2 D 2 D 2 B 3 D 3 B 3 A 3 A 3 D 3 C 4 D 4 C 4 C 4 E 4 E 4 A 5 B 5 A 5 E 5 C 5 C 5 C 6 C 6 E 6 C 6 E 6 A 6 A 7 E 7 C 7 D 7 C 7 B 7 D 8 C 8 D 8 A 8 D 8 D 8 D 9 A 9 E 9 B 9 D 9 D 9 E 10 B 10 B 10 B 10 D 10 D 10 C 11 D 11 E 11 D 11 C 11 C 11 A 12 D 12 C 12 B 12 C 12 B 12 D 13 D 13 D 13 D 13 B 13 C 13 A 14 D 14 B 14 C 14 C 14 D 14 C 15 B 15 D 15 D 15 C 15 B 15 C 16 B 16 C 16 D 16 D 16 E 16 D 17 A 17 D 17 C 17 C 17 B 17 E 18 E 18 C 18 D 18 E 18 D 18 B 19 A 19 D 19 A 19 D 19 B 19 C 20 C 20 B 20 C 20 B 20 D 20 E 21 D 21 E 21 B 21 E 21 A 21 E 22 E 22 D 22 D 22 D 22 D 22 C 23 D 23 E 23 D 23 B 23 A 23 D 24 C 24 C 24 B 24 D 24 C 24 B 25 D 25 D 25 C 25 D 26 B 26 E 26 C 26 D 27 D 27 B 27 C 27 A 28 D 28 C 28 C 28 B 29 C 29 C 29 B 29 E 30 B 30 A 30 D 30 B Answer of IKMC 2012 Page 1