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IKMC 2025 - Grades 9-10 - Self-study

Question 1

In which of the following hexagons is exactly one third of the area black and half of the area white?
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Question 2

The base of a triangle is extended by 50% and its height is reduced by one third. What is the ratio of the area of the new triangle to the area of the original triangle?
  1. 2:1
  2. 1:1
  3. 1:2
  4. 1:3
  5. 1:4

Question 3

The left and right parts of a three-part brochure each contain four transparent windows. When both parts are folded onto the middle part, some of its numbers are visible through the windows. What is the sum of the visible numbers when the brochure is folded?

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  1. 7
  2. 9
  3. 12
  4. 14
  5. 15

Question 4

Every year, the third Thursday in March is Kangaroo Day. What is the earliest calendar day that can be a Kangaroo Day?
  1. March 14th
  2. March 15th
  3. March 20th
  4. March 21st
  5. March 22nd

Question 5

A recipe needs $1\frac12$ cups of water for 1 cup of rice. Ruben wants to use $1\frac12$ cups of rice. How many cups of water does he need?
  1. 1
  2. $1\frac14$
  3. $1\frac34$
  4. $2\frac14$
  5. $2\frac12$

Question 6

Lisa has four wooden digits with which she can make the number 2025. How many different numbers greater than 2025 can she make using these four digits?

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  1. 3
  2. 6
  3. 8
  4. 9
  5. 11

Question 7

Sarah has a bag of 18 balls numbered from 1 to 18. What is the smallest number of balls she must remove to be sure that at least three removed balls bear prime numbers?
  1. 11
  2. 12
  3. 13
  4. 14
  5. 15

Question 8

Luka has dogs, rabbits and cats as pets. Eight pets are not dogs, five are not rabbits and seven are not cats. How many pets does Luka have?
  1. 10
  2. 11
  3. 15
  4. 16
  5. 20

Question 9

Katrin and Thomas both celebrate their birthday today. Thomas notices that $1/19$ of Katrin’s age equals $1/17$ of his age. Together they are older than 40 years and younger than 100 years. How old is Katrin?
  1. 19
  2. 34
  3. 38
  4. 57
  5. 76

Question 10

A circle has centre O and radius 10 cm. A square OPQR is drawn inside it so that Q lies on the circle, as shown. What is the area of the grey triangle PQR?

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  1. 12.5 cm²
  2. 25 cm²
  3. 50 cm²
  4. 75 cm²
  5. 100 cm²

Question 11

On a standard die, the numbers on opposite faces always sum to 7. We tilt the shown die repeatedly over its edges so that each of its six faces is on top once. Which sequence of top numbers is NOT possible?

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  1. 3-5-1-2-6-4
  2. 3-2-5-1-6-4
  3. 3-1-2-6-5-4
  4. 3-1-5-6-2-4
  5. 3-6-2-1-5-4

Question 12

Alexander folds a square sheet of paper along its diagonal to form a triangle. He then folds it again so that one shorter side lies on the longer side, producing the smaller triangle AXC shown. What is $\angle CXA$?

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  1. 108°
  2. 112.5°
  3. 120°
  4. 145°
  5. 157.5°

Question 13

The four-digit number $\overline{80ab}$ is missing its last two digits. It is divisible by both 8 and 9. What is the product of the two missing digits?
  1. 6
  2. 16
  3. 20
  4. 24
  5. 48

Question 14

An athlete has 2 gold medals and 5 silver medals, numbered in some order from 1 to 7. Each of the six black-and-white photographs shows exactly one gold medal. What is the sum of the numbers on the two gold medals?

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  1. 7
  2. 8
  3. 9
  4. 10
  5. 11

Question 15

Anna is looking at a picture on her smartphone. Its format is 16:9 and it fills the entire screen. When she turns the phone, the picture becomes smaller, as shown. What proportion of the screen does the smaller picture occupy?

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  1. $3/4$
  2. $9/16$
  3. $27/64$
  4. $32/81$
  5. $81/256$

Question 16

Paul shoots at the two targets shown a total of 27 times. He hits the upper-left target on 50% of the shots aimed at it and the bottom-right target on 80% of the shots aimed at it. Altogether 9 shots miss their target. How many times does he hit the upper-left target?

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  1. 4
  2. 5
  3. 6
  4. 7
  5. 8

Question 17

Some cards on a table bear distinct positive integers, all smaller than 20. Their product is 2025. What is the maximum possible number of cards?
  1. 2
  2. 3
  3. 4
  4. 5
  5. 6

Question 18

N is the largest six-digit number whose digits have product 180. What is the sum of the digits of N?
  1. 21
  2. 22
  3. 23
  4. 24
  5. 25

Question 19

Five bricks form the wall shown. Peter may remove a brick only when no other brick is directly above it. At each turn he chooses uniformly at random among the currently removable bricks and removes one. What is the probability that brick 4 is the third brick removed?

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  1. $1/3$
  2. $1/4$
  3. $1/5$
  4. $1/6$
  5. $1/8$

Question 20

Daniel numbers cells of squared paper 1, 2, 3, …, 2025 in the counterclockwise spiral shown, starting from a chosen cell. He considers the figure formed by all 2025 numbered cells. Each cell has side length 0.5 cm. What is the perimeter of the figure?

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  1. 25 cm
  2. 45 cm
  3. 80 cm
  4. 90 cm
  5. 180 cm