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IKMC 2019 - Grades 11-12 - Self-study

Question 1

The flag of Kangoraland is a rectangle which is split into three equal rectangles as shown. How big is the ratio of the side lengths of the white rectangle?

Question illustration
  1. 1:2
  2. 2:3
  3. 2:5
  4. 3:7
  5. 4:9

Question 2

The numbers 1,2,3,4 are placed in different cells of the $2\times2$ table. The two row sums and two column sums are calculated. Two of these sums are 4 and 5. What are the other two sums?

Question illustration
  1. 6 and 6
  2. 3 and 5
  3. 4 and 5
  4. 4 and 6
  5. 5 and 6

Question 3

A rectangle is coloured in five different ways as shown. In which picture is the grey area biggest?
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Question 4

Three triangles are connected to each other as shown. In which of the following pictures are the three triangles connected in the same way?

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Question 5

A pyramid has 23 triangular faces. How many edges does this pyramid have?
  1. 23
  2. 24
  3. 46
  4. 48
  5. 69

Question 6

Three four-digit numbers are written onto three separate pieces of paper as shown. The sum of the three numbers is 11126. Three of the digits in the picture are hidden. Which are the three hidden digits?

Question illustration
  1. 1,4,7
  2. 1,5,7
  3. 3,3,3
  4. 4,5,6
  5. 4,5,7

Question 7

Reading from the left, what is the first digit of the smallest positive integer whose digit sum is 2019?
  1. 2
  2. 3
  3. 4
  4. 5
  5. 6

Question 8

How many integers from $2^{10}$ to $2^{13}$, inclusive, are divisible by $2^{10}$?
  1. 2
  2. 4
  3. 6
  4. 8
  5. 16

Question 9

Each face of a fair die has 1,2 or 3 dots. The probabilities of rolling 1,2,3 are respectively $1/2,1/3,1/6$. Which picture cannot show this die?
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Question 10

Every day the three kangaroos Alex, Bob and Carl go for a walk. If Alex does not wear a hat, Bob wears a hat. If Bob does not wear a hat, Carl wears a hat. Today Carl does not wear a hat. Which kangaroos can one say for sure are wearing a hat today?
  1. Only Alex and Bob
  2. Only Alex
  3. Alex, Bob and Carl
  4. Neither Alex nor Bob
  5. Only Bob

Question 11

What is the highest power of 3 dividing $7!+8!+9!$? Here $n!=n(n-1)\cdots2\cdot1$.
  1. $3^2$
  2. $3^4$
  3. $3^5$
  4. $3^6$
  5. A power greater than $3^6$

Question 12

In this school year the number of boys in my class has increased by 20% compared to the previous year and the number of girls has decreased by 20%. There is now one person more than before in this class. Which of the following numbers could be the current number of students in my class?
  1. 22
  2. 26
  3. 29
  4. 31
  5. 34

Question 13

A cuboid container, not completely full, contains $120\,\mathrm{m}^3$ of water. The depth is 2 m, 3 m or 5 m depending on which face is on the ground, as shown (not to scale). What is the container’s volume?

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  1. $160\,\mathrm{m}^3$
  2. $180\,\mathrm{m}^3$
  3. $200\,\mathrm{m}^3$
  4. $220\,\mathrm{m}^3$
  5. $240\,\mathrm{m}^3$

Question 14

Michael defines $x\diamond y=y-x$ for real numbers. If $(a\diamond b)\diamond c=a\diamond(b\diamond c)$, which statement must be true?
  1. $a=b$
  2. $b=c$
  3. $a=c$
  4. $a=0$
  5. $c=0$

Question 15

Three pulleys are connected by two ropes as shown. The free end P is pulled down 24 cm. By how many centimetres does Q move up?

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  1. 24
  2. 12
  3. 8
  4. 6
  5. $\frac{24}5$

Question 16

A positive integer $n$ is called good if its greatest positive divisor other than itself equals $n-6$. How many good integers are there?
  1. 1
  2. 2
  3. 3
  4. 6
  5. Infinitely many

Question 17

There are five balls in a box. Four of which contain chocolate, one contains one boiled sweet. Johann and Maria take it in turns to draw a ball from the box without replacing it. Whoever draws the boiled sweet wins. Johann starts. How big is the probability that Maria wins?
  1. $\frac25$
  2. $\frac35$
  3. $\frac12$
  4. $\frac56$
  5. $\frac13$

Question 18

The diagram shows adjoining squares of side lengths $a,b$, with $a\lt b$. What is the grey triangle’s area?

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  1. $\sqrt{ab}$
  2. $\frac12a^2$
  3. $\frac12b^2$
  4. $\frac14(a^2+b^2)$
  5. $\frac12(a^2+b^2)$

Question 19

What is the greatest integer smaller than $\sqrt{20+\sqrt{20+\sqrt{20+\sqrt{20+\sqrt{20}}}}}$?
  1. 4
  2. 5
  3. 6
  4. 20
  5. 25

Question 20

The ten vertices of the three-square network shown are labelled 1 to 10, each once. The sum S of the four corner labels is the same for every square. What is the smallest S?

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  1. 18
  2. 19
  3. 20
  4. 21
  5. 22