kangaroo 2015 past papers cadet level -...

Muhammad Javed Iqbal

Kangaroo 2015 Past Papers Cadet Level

Kangaroo 2005— Cadet Max Time: 75 min

1

Cadet 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)31 (B)

21 (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 (B) 54 cm (C) 60 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) 60

6. 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) − a2 (C) 3a(b − a) + a2

(D) 3ab


2

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) 50 8. 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) 65 10. 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) 1002 12. 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.14.40 (B) Rs.14.60 (C) Rs.14.80 (D) Rs.15.20 13. 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


3

B

A D

CC

75°

50° ?

30°

14. Some angles in quadrilateral ABCD are shown in the figure. If ADBC , then what is the angle ADC? (A) 75° (B) 50° (C) 55° (D) 65° 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? (A) 31 (B) 12 (C) 34 (D) 7 16. 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) 136° 18. 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


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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) 1010 20. 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 cm2 (B) 12 cm2

(C) 14 cm2 (D) 16 cm2

22. 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 = 2 · 3 · 3 · 5. 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) 720

25. 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

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


Cadet: Class (7-8)

3-Point-Problems 1. The contest Kangaroo in Europe has taken place every year since 1991. So, the contest

Kangaroo in 2006 is the A) 15th B) 16 th C) 17 th D) 14 th

2. 20(0+6) – (200)+6 =

A) 106 B) 114 C) 126 D) 12 3. 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%

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, 1200 - 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

O


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 cm2 B) 58 cm2 C) 59 cm2 D) 60 cm2

8. A container that can hold 12 litres is 43 full. How much will it contain after 4 litres has been

poured out of it? A) 5 B) 6 C) 8 D) 12

9. 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) 27

10. An air temperature fell 10o 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. A) It’s red and round It’s either blue or yellow. B) It’s a blue square If it’s yellow, it’s square. C) It’s blue and round It’s either square or round. D) It’s yellow and round

12. Three Tuesdays of a month fall on even dates. What day of a week was the 21st 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) 40

15. 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

Kangaroo 2006— Cadet Max Time: 95 min 16. Which number when squared is increased by 500%?

A) 5 B) 6 C) 7 D) 10 17. 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) 12

18. 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) 4

19. 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) 100

5-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) 48

22. 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) 256

23. 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 0.001 C) equal to 25 D) equal to 100


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) 334

25. 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


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

INTERNATIONAL KANGAROO MATHEMATICS CONTEST 2007

Level Cadet: Class (7 & 8) Max Time: 1 Hour & 35 Min

3-Point-Problems

Q1. 2007

2 0 0 7+ + +=

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

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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 13931 is a palindromic number. What is the difference between the smallest 5-digit palindromic number and the largest 6-digit palindromic numbers?

A) 989989 B) 989998 C) 998998 D) 999898

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 AA1A2 . . . A12B (see the Fig.). Find the length of AA1A2 . . . A12B.

A) 48 cm B) 72 cm C) 96 cm D) 106 cm

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INTERNATIONAL KANGAROO MATHEMATICS CONTEST 2007Q12. On parallel lines l1 and l2, 6 points were drawn; 4 on line l1 and 2 on line l2. 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 BB) 7/12 of the customers buy product A, 5/12 buy product BC) 1/2 of the customers buy product A, 1/2 buy product BD) 1/3 of the customers buy product A, 2/3 buy product B

Q14. In order to obtain the number 88, we must raise 44 to the power

A) 3 B) 2 C) 4 D) 8

Q15. ABC and CDE are equal equilateral triangles. If angle80ACD = o , what is angle ABD?

A) 25o B) 30o C) 35o D) 40o

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

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INTERNATIONAL KANGAROO MATHEMATICS CONTEST 2007Q19. 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=90o and ∠ COD=90o (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) 2.25

5-Point-ProblemsQ21. 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


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

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?

A) 3 B) 4 C) 5 D) 6

2)

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?

A) 0 B) 1 C) 2 D) 3

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)? A) 24 cm B) 28 cm C) 32 cm D) It depends upon the triangle measures

4cm

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) 56

9) 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 х2. He is known to have died in 1899. When was he born?

A) 1806 B) 1848 C) 1849 D) another answer

124° 108°

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? 1 2 3 4 5

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?


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?

A) 27 cm2 B) 45 cm2 C) 54 cm2 D) 108 cm2

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

P

R

Q

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 113 unit cubes. What is the largest number of unit cubes visible from a same point of view?

A) 329 B) 330 C) 331 D) 332

5-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) 785

22) 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 !

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

Inernational Mathematics Contest 2008Answer of Questions

Answer of IKMC 2008

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

International Kangaroo Mathematics Contest 2009Answer of Questions

Answer of IKMC 2009.xls

1

International Kangaroo Mathematics Contest 2010 Cadet Level: Class (7 & 8) Max Time: 2 Hours

3-point problems

Q1) How much is 12 + 23 + 34 + 45 + 56 + 67 + 78 + 89?

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

2

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) 10100

4-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 one-digit 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.

3

Q15) Saima has wound some rope around a piece of wood. She rotates the wood as shown with the arrow.

Front side

What does she see after the rotation?

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?


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

4

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 1 × 1 × 1. Each cube is a single colour. Kangaroo wants to use 27 small cubes to make a 3 × 3 × 3 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

Q. No. Ecolier Level Q. No. Benjamin Level Q. No. Cadet Level Q. No. Junior Level Q. No. Student Level1 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


Level Cadet – Class 7 & 8

3 point problems PROBLEM 01 Which of the following has the largest value? (A) 2011 (B) 1 (C) 1 × 2011 (D) 1 + 2011 (E) 1 ÷ 2011 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 cm2. What is the difference between the area of the medium square and the area of the large square? (A) 6 cm2 (B) 9 cm2 (C) 12 cm2 (D) 15 cm2 (E) 18 cm2


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


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 cm2 (B) 64 cm2 (C) 110.25 cm2 (D) 144 cm2 (E) 256 cm2


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 퐷퐸 of length 2 cm on a piece of paper. How many different points F can she draw on the paper so that the triangle 퐷퐸퐹 is right-angled and has area 1 cm2? (A) 2 (B) 4 (C) 6 (D) 8 (E) 10 PROBLEM 19 The positive number 푎 is less than 1, and the number 푏 is greater then 1. Which of the following numbers has the largest value? (A) 푎 × 푏 (B) 푎 + 푏 (C) 푎 ÷ 푏 (D) b (E) the answer depends on 푎 and 푏. 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 `24푋8푌' is divisible by 4, 5 and 9. What is the sum of the digits 푋 and 푌? (A) 13 (B) 10 (C) 9 (D) 5 (E) 4


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)

(C)

(D) (E)

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 cm2 (B) 10 cm2 (C) 11 cm2 (D) 15 cm2 (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 퐴퐵퐶퐷 with 퐴퐵 = 퐴퐶 , the following angles are known: ∡ 퐵퐴퐷 = 80°, ∡ 퐴퐵퐶 = 75, ∡ 퐴퐷퐶 = 65°. What is the size of ∡ 퐵퐷퐶? (A) 10° (B) 15° (C) 20° (D) 30° (E) 45°


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 푥? (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

Q. No. Ecolier Level Q. No. Benjamin Level Q. No. Cadet Level Q. No. Junior Level Q. No. Student Level1 C 1 C 1 D 1 B 1 A2 C 2 C 2 A 2 C 2 B3 B 3 A 3 B 3 D 3 A4 E 4 B 4 A 4 A 4 C5 B 5 E 5 C 5 E 5 D6 A 6 E 6 C 6 C 6 C7 D 7 D 7 E 7 D 7 C8 C 8 A 8 A 8 C 8 C9 B 9 E 9 B 9 C 9 C

10 B 10 B 10 C 10 E 10 B11 B 11 D 11 C 11 B 11 D12 E 12 B 12 C 12 B 12 E13 D 13 E 13 B 13 A 13 D14 A 14 C 14 A 14 B 14 D15 C 15 D 15 E 15 C 15 B16 B 16 B 16 D 16 D 16 A17 C 17 B 17 B 17 D 17 E18 C 18 C 18 C 18 D 18 D19 C 19 D 19 B 19 C 19 A20 E 20 E 20 A 20 B 20 B21 D 21 C 21 E 21 C 21 B22 A 22 C 22 D 22 C 22 C23 E 23 D 23 B 23 B 23 A24 D 24 C 24 D 24 C 24 A

25 D 25 D 25 B 25 D26 E 26 D 26 D 26 B27 D 27 A 27 B 27 C28 A 28 B 28 C 28 D29 D 29 B 29 C 29 C30 A 30 B 30 E 30 C


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 onechocolate bar?

(A) 1 EUR (B) 2 EUR (C) 3 EUR (D) 4 EUR (E) 5 EUR

2. 11.11 − 1.111 =

(A) 9.009 (B) 9.0909 (C) 9.99 (D) 9.999 (E) 10

3. A watch is placed face up on a table so that its minute hand points north-east. How manyminutes pass before the minute hand points north-west for the first time?

(A) 45 (B) 40 (C) 30 (D) 20 (E) 15

4. 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 fallsapart 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 sixheads are chopped off one by one, how many heads will the dragon finally have?

(A) 25 (B) 28 (C) 29 (D) 30 (E) 35

6. In which of the following expressions can we replace each occurrence of the number 8 bythe same positive number (other than 8) and obtain the same result?

(A) (8 + 8) : 8 + 8 (B) 8 · (8 + 8) : 8 (C) 8 + 8 − 8 + 8

(D) (8 + 8 − 8) · 8 (E) (8 + 8 − 8) : 8

7. Each of the nine paths in a park is 100 m long. Ann wants to go from A to B withoutgoing along any path more than once. What is the length of the longest route she can choose?

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A

B

(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 canyou choose two vertices, one in each triangle, so that the straight line through the vertices doesnot 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 apair of scissors. He then opens up the paper again. Which of the following shapes cannot bethe 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) (C)

(D) (E)

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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 and8 exactly once. Kanga wants the sum of the two numbers to be as small as possible. What isthe value of this smallest possible sum?

(A) 2468 (B) 3333 (C) 3825 (D) 4734 (E) 6912

12. Mrs Gardner grows peas and strawberries. This year she has changed the rectangularpea bed to a square by lengthening one of its sides by 3 metres. As a result of this change, thearea of the strawberry bed was reduced by 15 m2. What was the area of the pea bed before

the change?

Strawberries

Peas

Last year

Strawberries

Peas

This year

(A) 5 m2 (B) 9 m2 (C) 10 m2 (D) 15 m2 (E) 18 m2

13. Barbara wants to complete the diagram by inserting three numbers, one in each emptycell. She wants the sum of the first three numbers to be 100, the sum of the three middlenumbers to be 200 and the sum of the last three numbers to be 300. What number shouldBarbara insert in the middle cell of the diagram?

10 130

(A) 50 (B) 60 (C) 70 (D) 75 (E) 100

14. In the figure, what is the value of x?

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58◦

100◦93◦

x◦

(A) 35 (B) 42 (C) 51 (D) 65 (E) 109

15. 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 fournumbers are 2, 5, 7 and 12. On each card, the number does not correspond to the phrase onthe 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 largerequilateral triangle with sides of 6 cm, as shown.

The sum of the perimeters of the three small triangles is equal to the perimeter of theremaining 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, anumber of mice came and stole some pieces, watched by the lazy cat Ginger. Ginger noticedthat each mouse stole a different number of pieces less than 10, and that no mouse stole exactlytwice as many pieces as any other mouse. What is the largest number of mice that Ginger

4 of 7


could have seen stealing cheese?

(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

18. At the airport there is a moving walkway 500 metres long, which moves with a speed of4 km/hour. Ann and Bill step on the walkway at the same time. Ann walks with a speed of 6km/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 hissides 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 squareafter the four statements?

(A) 28 (B) 80 (C) 88 (D) 112 (E) 120

20. A cube is rolled on a plane so that it turns around its edges. Its bottom face passesthrough 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? 1 2 3

4 5

6 7

(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 differencebetween the heights of any two neighbouring cubes is 2 cm. The largest cube is as high as atower 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

to AC. A B

CD

M

N

What is the ratio of the area of the shaded triangle MNC

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to the area of the square?

(A) 1:6 (B) 1:5 (C) 7:36 (D) 3:16 (E) 7:40

23. The tango is danced in pairs, each consisting of one man and one woman. At a danceevening no more than 50 people are present. At one moment 3/4 of the men are dancing with4/5 of the women. How many people are dancing at that moment?

(A) 20 (B) 24 (C) 30 (D) 32 (E) 46

24. David wants to arrange the twelve numbers from 1 to 12 in a circle so that any twoneighbouring numbers differ by either 2 or 3. Which of the following pairs of numbers have tobe neighbours?

(A) 5 and 8 (B) 3 and 5 (C) 7 and 9 (D) 6 and 8 (E) 4 and 6

25. Some three-digit integers have the following property: if you remove the first digit of thenumber, you get a perfect square; if instead you remove the last digit of the number, you alsoget 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) 2016

26. 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 storiesthat can start on an odd-numbered page?

(A) 15 (B) 18 (C) 20 (D) 21 (E) 23

27. An equilateral triangle starts in a given position and is moved to new positions in asequence 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 (3n)◦). How manydifferent positions, including the initial position, will the triangle occupy? Two positions areconsidered equal if the triangle covers the same part of the plane.

(A) 3 (B) 4 (C) 5 (D) 6 (E) 360

28. A rope is folded in half, then in half again, and then in half again. Finally the foldedrope is cut through, forming several strands. The lengths of two of the strands are 4 m and 9m. 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

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29. A triangle is divided into four triangles and three quadrilaterals by three straight linesegments. The sum of the perimeters of the three quadrilaterals is equal to 25 cm. The sum ofthe perimeters of the four triangles is equal to 20 cm. The perimeter of the whole triangle isequal 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) 16

30. A positive number is to be placed in each cell of the 3 × 3 grid shown, so that: in eachrow 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. What number should be placed inthe central cell?

(A) 16 (B) 8 (C) 4 (D) 14

(E) 18

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Q. No. Pre-Ecolier Level Q. No. Ecolier Level Q. No. Benjamin Level Q. No. Cadet Level Q. No. Junior Level Q. No. Student Level1 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

KSF 2013 – finalized problems Cadet

3 points

# 1. In the picture, the big triangle is equilateral and has area 9. The lines are parallel to the sidesand divide the sides into three equal parts. What is the area of the shaded part?

(A) 1 (B) 4 (C) 5 (D) 6 (E) 7

# 2. It is true that 1111101 = 11. What is the value of 3333

101 + 6666303 ?

(A) 5 (B) 9 (C) 11 (D) 55

(E) 99

# 3. The masses of salt and fresh water in sea water in Protaras are in the ratio 7 : 193. How manykilograms of salt are there in 1000 kg of sea water?

(A) 35 (B) 186 (C) 193 (D) 200 (E) 350

# 4. Ann has the square sheet of paper shown on the left. By cutting along the lines of the square,she cuts out copies of the shape shown on the right. What is the smallest possible number of cells

remaining?

(A) 0 (B) 2 (C) 4 (D) 6 (E) 8

# 5. Roo wants to tell Kanga a number with the product of its digits equal to 24. What is the sumof the digits of the smallest number that Roo could tell Kanga?

(A) 6 (B) 8 (C) 9 (D) 10 (E) 11

# 6. A bag contains balls of five different colours. Two are red, three are blue, ten are white, fourare green and three are black. Balls are taken from the bag without looking, and not returned. Whatis the smallest number of balls that should be taken from the bag to be sure that two balls of thesame colour have been taken?

(A) 2 (B) 12 (C) 10 (D) 5 (E) 6

# 7. Alex lights a candle every ten minutes. Each candle burns for 40 minutes and then goes out.How many candles are alight 55 minutes after Alex lit the first candle?

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

1


# 8. The average number of children in five families cannot be

(A) 0.2 (B) 1.2 (C) 2.2 (D) 2.4 (E) 2.5

# 9. Mark and Liza stand on opposite sides of a circular fountain. They then start to run clockwiseround the fountain. Mark’s speed is 9/8 of Liza’s speed. How many circuits has Liza completed whenMark catches up with her for the first time?

(A) 4 (B) 8 (C) 9 (D) 2

(E) 72

# 10. The positive integers x, y and z satisfy x × y = 14, y × z = 10 and z × x = 35. What is thevalue of x+ y + z?

(A) 10 (B) 12 (C) 14 (D) 16 (E) 18

4 points

# 11. Carina and a friend are playing a game of ”battleships” on a 5× 5 board. Carina has alreadyplaced two ships as shown. She still has to place a 3 × 1 ship so that it covers exactly three cells. Notwo ships can have a point in common. How many positions are there for her 3 × 1 ship?

(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

# 12. In the diagram, α = 55◦, β = 40◦ and γ = 35◦. What is the value of δ?

α β

γ

δ

(A) 100◦ (B) 105◦ (C) 120◦ (D) 125◦ (E) 130◦

# 13. The perimeter of a trapezium is 5 and the lengths of its sides are integers. What are thesmallest two angles of the trapezium?

(A) 30◦ and 30◦ (B) 60◦ and 60◦ (C) 45◦ and 45◦

(D) 30◦ and 60◦ (E) 45◦ and 90◦

2


# 14. One of the following nets cannot be folded to form a cube. Which one?

(A) (B) (C) (D) (E)

# 15. Vasya wrote down several consecutive integers. Which of the following could not be thepercentage of odd numbers among them?

(A) 40 (B) 45 (C) 48 (D) 50 (E) 60

# 16. The edges of rectangle ABCD are parallel to the coordinate-axes. ABCD lies below the x-axisand to the right of the y-axis, as shown in the figure. The coordinates of the four points A, B, Cand D are all integers. For each of these points we calculate the value y-coordinate ÷ x-coordinate.Which of the four points gives the least value?

y

x

A B

CD

(A) A (B) B (C) C (D) D

(E) It depends on the rectangle.

# 17. All 4-digit positive integers with the same four digits as in the number 2013 are written on theblackboard in an increasing order. What is the largest possible difference between two neighbouringnumbers on the blackboard?

(A) 702 (B) 703 (C) 693 (D) 793 (E) 198

# 18. In the 6 × 8 grid shown, 24 of the cells are not intersected by either diagonal.

When the diagonals of a 6× 10 grid are drawn, how many of the cells are not intersected by eitherdiagonal?

(A) 28 (B) 29 (C) 30 (D) 31 (E) 32

# 19. 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. Also,Betty and Cathie were born in the same month. Andy and Cathie were born on the same day ofdifferent months. Also, Dannie and Eddy were born on the same day of different months. Which of

3


these children is the youngest?

(A) Andy (B) Betty (C) Cathie (D) Dannie (E) Eddy

# 20.

4 2 3 2

3 3 1 2

2 1 3 1

1 2 1 2

Back

Front John has made a building of cubes standing on a 4 × 4 grid. The diagramshows the number of cubes standing on each cell. When John looks from the back, what does he see?

(A) (B) (C) (D) (E)

5 points

# 21. The diagram shows a shaded quadrilateral KLMN drawn on a grid. Each cell of the grid hassides of length 2 cm. What is the area of KLMN?

KL

MN

(A) 96 cm2 (B) 84 cm2 (C) 76 cm2 (D) 88 cm2 (E) 104 cm2

# 22. Let S be the number of squares among the integers from 1 to 20136. Let Q be the number ofcubes among the same integers. Then

(A) S = Q (B) 2S = 3Q (C) 3S = 2Q (D) S = 2013Q (E) S3 = Q2

# 23. John chooses a 5-digit positive integer and deletes one of its digits to make a 4-digit number.The sum of this 4-digit number and the original 5-digit number is 52713. What is the sum of thedigits of the original 5-digit number?

(A) 26 (B) 20 (C) 23 (D) 19 (E) 17

# 24. A gardener wants to plant twenty trees (maples and lindens) along an avenue in the park. Thenumber of trees between any two maples must not be equal to three. Of these twenty trees, what isthe greatest number of maples that the gardener can plant?

(A) 8 (B) 10 (C) 12 (D) 14

(E) 16

4


# 25. Andrew and Daniel recently took part in a marathon. After they had finished, they noticedthat Andrew finished ahead of twice as many runners as finished ahead of Daniel, and that Danielfinished ahead of 1.5 times as many runners as finished ahead of Andrew. Andrew finished in 21stplace. How many runners took part in the marathon?

(A) 31 (B) 41 (C) 51 (D) 61 (E) 81

# 26. Four cars enter a roundabout at the same time, each one from a different direction, as shownin the diagram. Each of the cars drives less than once round the roundabout, and no two cars leavethe roundabout in the same direction. How many different ways are there for the cars to leave theroundabout?

(A) 9 (B) 12 (C) 15 (D) 24 (E) 81

# 27. A sequence starts 1, −1, −1, 1, −1. After the fifth term, every term is equal to the productof the two preceding terms. For example, the sixth term is equal to the product of the fourth termand the fifth term. What is the sum of the first 2013 terms?

(A) −1006 (B) −671 (C) 0 (D) 671 (E) 1007

# 28. Ria bakes six raspberry pies one after the other, numbering them 1 to 6 in order, with thefirst being number 1. Whilst she is doing this, her children sometimes run into the kitchen and eatthe hottest pie. Which of the following could not be the order in which the pies are eaten?

(A) 123456 (B) 125436 (C) 325461 (D) 456231 (E) 654321

# 29. Each of the four vertices and six edges of a tetrahedron is marked with one of the ten numbers1, 2, 3, 4, 5, 6, 7, 8, 9 and 11 (number 10 is omitted). Each number is used exactly once. For anytwo vertices of the tetrahedron, the sum of two numbers at these vertices is equal to the number onthe edge connecting these two vertices. The edge PQ is marked with the number 9. Which number is

used to mark edge RS?

P

Q

R

S

9

?

(A) 4 (B) 5 (C) 6 (D) 8 (E) 11

# 30. A positive integer N is smaller than the sum of its three greatest divisors (naturally, excludingN itself). Which of the following statements is true?

(A) All such N are divisible by 4.(B) All such N are divisible by 5.(C) All such N are divisible by 6.

(D) All such N are divisible by 7.(E) There is no such N .

5

Q. No. Pre-Ecolier Level Q. No. Ecolier Level Q. No. Benjamin Level Q. No. Cadet Level Q. No. Junior Level Q. No. Student Level1 D 1 D 1 E 1 D 1 D 1 C

2 B 2 D 2 C 2 D 2 C 2 C

3 A 3 E 3 C 3 A 3 C 3 E

4 D 4 B 4 B 4 C 4 C 4 D

5 B 5 C 5 E 5 E 5 C 5 C

6 E 6 D 6 B 6 E 6 E 6 D

7 C 7 B 7 B 7 C 7 E 7 E

8 D 8 E 8 E 8 E 8 C 8 B

9 E 9 B 9 C 9 A 9 D 9 E

10 C 10 D 10 C 10 C 10 C 10 D

11 A 11 D 11 C 11 E 11 D 11 C

12 B 12 A 12 C 12 E 12 B 12 A

13 D 13 D 13 D 13 B 13 D 13 E

14 E 14 E 14 B 14 C 14 D 14 D

15 C 15 B 15 E 15 B 15 D 15 A

16 A 16 B 16 B 16 A 16 D 16 D

17 E 17 D 17 D 17 A 17 A 17 A

18 C 18 D 18 A 18 E 18 C 18 E

19 C 19 B 19 C 19 B 19 D 19 E

20 B 20 C 20 D 20 C 20 E 20 C

21 A 21 E 21 A 21 B 21 C 21 A

22 D 22 B 22 D 22 D 22 D 22 D

23 D 23 B 23 B 23 C 23 C 23 A

24 D 24 B 24 A 24 C 24 E 24 E

25 D 25 B 25 B 25 D

26 D 26 A 26 D 26 C

27 B 27 B 27 C 27 B

28 B 28 D 28 C 28 D

29 D 29 B 29 C 29 E

30 B 30 C 30 B 30 B


Answer of IKMC 2013.xls Page 1


3 points

# 1. Each year, the date of the Kangaroo competition is the third Thursday of March. What is thelatest possible date of the competition in any year?

(A) 14th March (B) 15th March (C) 20th March (D) 21st March (E) 22nd March

# 2. How many quadrilaterals of any size are shown in the figure?

(A) 0 (B) 1 (C) 2 (D) 4 (E) 5

# 3. What is the result of: 2014 · 2014 : 2014 − 2014?

(A) 0 (B) 1 (C) 2013 (D) 2014 (E) 4028

# 4. 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?

A B

CD

M N

(A) 0,5 (B) 5 (C) 2,5 (D) 7.5 (E) 10

# 5. 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. Wanda has several square pieces of paper of area 4. She cuts them into squares and right-angledtriangles in the manner shown in the first diagram. She takes some of the pieces and makes the birdshown in the second diagram. What is the area of the bird?

(A) 3 (B) 4 (C) 9/2 (D) 5 (E) 6

# 7. 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

1


# 8. 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. Which of the following calculations gives the largest result?

(A) 44 × 777 (B) 55 × 666 (C) 77 × 444

(D) 88 × 333 (E) 99 × 222

# 10. 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 thatArno can take?

(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

4 points

# 11. Jack has a piano lesson twice a week and Hannah has a piano lesson every other week. In agiven 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. In the diagram, the area of each circle is 1cm2. The area common to two overlapping circlesis 1

8cm2. What is the area of the region covered by the five circles?

(A) 4cm2 (B) 92cm

2 (C) 358 cm

2 (D) 398 cm

2 (E) 194 cm

2

# 13. This year a grandmother, her daughter and her granddaughter noticed that the sum of theirages 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. Five equal rectangles are placed inside a square with side 24 cm, as shown in the diagram.

2


What is the area of one rectangle?

(A) 12 cm2 (B) 16 cm2 (C) 18 cm2 (D) 24 cm2 (E) 32 cm2

# 15. The heart and the arrow are in the positions shown in the figure. At the same time theheart and the arrow start moving. The arrow moves three places clockwise and the heart moves fourplaces anticlockwise and then stop. They continue the same routine over and over again. After howmany 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. The diagram shows the triangle ABC in which BH is a perpendicular height and AD is theangle 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. Six boys share a flat with two bathrooms which they use every morning beginning at 7:00o’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 thatthey can finish using the bathrooms?

(A) 7:45 (B) 7:46 (C) 7:47 (D) 7:48 (E) 7:50

# 18. A rectangle has sides of length 6 cm and 11 cm. One long side is selected. The bisectors ofthe angles at either end of that side are drawn. These bisectors divide the other long side into threeparts. 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. Captain Sparrow and his pirate crew dug up several gold coins. They divide the coins amongstthemselves so that each person gets the same number of coins. If there were four fewer pirates, then

3


each person would get 10 more coins. However, if there were 50 fewer coins, then each person wouldget 5 fewer coins. How many coins dig they dig up?

(A) 80 (B) 100 (C) 120 (D) 150 (E) 250

# 20. The average of two positive numbers is 30% less than one of them. By what percentage is theaverage greater than the other number?

(A) 75% (B) 70% (C) 30% (D) 25% (E) 20%

5 points

# 21. Andy enters all the digits from 1 to 9 in the cells of a 3x3 table, so that each cell contains onedigit. 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. An antique scale is not working properly. If something is lighter than 1000 g, the scale showsthe correct weight. However, if something is heavier than or equal to 1000 g, the scale can show anynumber above 1000 g. We have 5 weights A g, B g, C g, D g, E g each under 1000 g. When theyare 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. Quadrilateral ABCD has right angles only at vertices A and D. The numbers show the areasof two of the triangles. What is the area of ABCD?

A B

CD

105

(A) 60 (B) 45 (C) 40 (D) 35 (E) 30

# 24. 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 points. How manyproblems were solved by both of them?

(A) 53 (B) 54 (C) 55 (D) 56 (E) 57

# 25. David rides his bicycle from Edinburgh to his croft. He was going to arrive at 15:00, buthe spent 2/3 of the planned time covering 3/4 of the distance. After that, he rode more slowly andarrived exactly on time. What is the ratio of the speed for the first part of the journey to the speed

4


for the second part?

(A) 5 : 4 (B) 4 : 3 (C) 3 : 2 (D) 2 : 1 (E) 3 : 1

# 26. We have four identical cubes (see picture). They are arranged so that a big black circle appearson one face, as shown in the second picture. What can be seen on the opposite face?

(A) (B) (C) (D) (E)

# 27. 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 ofthem 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. Several different positive integers are written on the board. Exactly two of them are divisibleby 2 and exactly 13 of them are divisible by 13. Let M be the greatest of these numbers. What is thesmallest possible value of M?

(A) 169 (B) 260 (C) 273 (D) 299 (E) 325

# 29. On a pond there are 16 water lily leaves in a 4 by 4 pattern as shown. A frog sits on a leafin one of the corners. It then jumps from one leaf to another either horizontally or vertically. Thefrog always jumps over at least one leaf and never lands on the same leaf twice. What is the greatestnumber of leaves (including the one it sits on) that the frog can reach?

(A) 16 (B) 15 (C) 14 (D) 13 (E) 12

# 30. A 5×5 square is made from 1×1 tiles, all with the same pattern, as shown. Any two adjacenttiles have the same colour along the shared edge. The perimeter of the large square consists of greyand white segments of length 1. What is the smallest possible number of such unit grey segments?

(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

5

Q. No. Pre-Ecolier Level Q. No. Ecolier Level Q. No. Benjamin Level Q. No. Cadet Level Q. No. Junior Level Q. No. Student Level1 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 B 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




1

Cadet 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)31 (B)

21 (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 (B) 54 cm (C) 60 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) 60

6. 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) − a2 (C) 3a(b − a) + a2

(D) 3ab


2

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) 50 8. 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) 65 10. 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) 1002 12. 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.14.40 (B) Rs.14.60 (C) Rs.14.80 (D) Rs.15.20 13. 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


3

B

A D

CC

75°

50° ?

30°

14. Some angles in quadrilateral ABCD are shown in the figure. If ADBC , then what is the angle ADC? (A) 75° (B) 50° (C) 55° (D) 65° 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? (A) 31 (B) 12 (C) 34 (D) 7 16. 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) 136° 18. 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


4

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) 1010 20. 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 cm2 (B) 12 cm2

(C) 14 cm2 (D) 16 cm2

22. 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 = 2 · 3 · 3 · 5. 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) 720

25. 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

Q. No. Pre-Ecolier Level Q. No. Ecolier Level Q. No. Benjamin Level Q. No. Cadet Level Q. No. Junior Level Q. No. Student Level1 D 1 E 1 B 1 E 1 B 1 E

2 C 2 A 2 C 2 B 2 E 2 A

3 B 3 E 3 A 3 E 3 B 3 A

4 C 4 E 4 A 4 A 4 E 4 A

5 B 5 B 5 B 5 D 5 B 5 D

6 E 6 E 6 E 6 A 6 C 6 D

7 B 7 B 7 D 7 C 7 E 7 B

8 E 8 A 8 E 8 D 8 A 8 E

9 C 9 D 9 A 9 C 9 D 9 C

10 B 10 B 10 A 10 D 10 C 10 B

11 C 11 C 11 D 11 B 11 B 11 C

12 C 12 C 12 E 12 C 12 C 12 D

13 A 13 B 13 C 13 B 13 C 13 E

14 E 14 D 14 C 14 D 14 A 14 C

15 D 15 C 15 D 15 E 15 B 15 B

16 E 16 A 16 C 16 C 16 D 16 A

17 D 17 C 17 B 17 C 17 D 17 C

18 D 18 D 18 C 18 B 18 B 18 C

19 D 19 D 19 E 19 A 19 D 19 C

20 C 20 E 20 B 20 B 20 E 20 A

21 C 21 C 21 E 21 C 21 A 21 D

22 A 22 D 22 C 22 C 22 B 22 A

23 A 23 D 23 D 23 D 23 C 23 D

24 D 24 C 24 C 24 A 24 D 24 C

25 B 25 D 25 C 25 E

26 C 26 D 26 A 26 D

27 B 27 E 27 B 27 C

28 D 28 C 28 D 28 B

29 E 29 C 29 B 29 D

30 D 30 D 30 B 30 D



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