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IKMC 2024 - Grades 9-10 - Self-study

Question 1

Which square is split into two parts that do not have the same shape?
  1. Question illustration
  2. Question illustration
  3. Question illustration
  4. Question illustration
  5. Question illustration

Question 2

Calculate $\frac{2\times0.24}{20\times2.4}$.
  1. 0.01
  2. 0.1
  3. 1
  4. 10
  5. 100

Question 3

Opposite faces of a die sum to 7. A vertex sum is the sum of the three faces meeting there; P has sum 1+2+3=6. What is the largest vertex sum at Q, R and S?

Question illustration
  1. 7
  2. 9
  3. 10
  4. 11
  5. 15

Question 4

Maya jumps into 48 squares, starting with her left foot, then both feet, right foot, both feet, and repeating. How many times does her left foot touch the floor?

Question illustration
  1. 12
  2. 24
  3. 36
  4. 40
  5. 48

Question 5

Tim draws every segment of the figure without lifting his pencil, possibly retracing segments. Segment lengths are shown. He may start anywhere. What is the minimum total drawn length?

Question illustration
  1. 14 cm
  2. 15 cm
  3. 16 cm
  4. 17 cm
  5. 18 cm

Question 6

Four equal touching circles lie in a square as shown. What is the ratio of the black area to the grey area?

Question illustration
  1. 1:4
  2. 1:3
  3. 2:3
  4. 3:4
  5. $1:\pi$

Question 7

For each a, b in {1,2,3,4,5,6}, the line y=ax+b forms a triangle with the coordinate axes. How many ordered pairs (a, b) give an isosceles triangle?
  1. 1
  2. 2
  3. 3
  4. 6
  5. 12

Question 8

John places a dark cube on a table, then covers its five exposed faces with five light cubes. What is the fewest additional dark cubes needed to cover every exposed light face?

Question illustration
  1. 8
  2. 9
  3. 10
  4. 13
  5. 19

Question 9

ABCD is a square with centre O. A regular hexagon has OC as one side, as shown. Find angle α.

Question illustration
  1. 105°
  2. 110°
  3. 115°
  4. 120°
  5. 125°

Question 10

A rectangle has perimeter 40 m and prime-number side lengths in metres. What is its largest possible area?
  1. 84 m²
  2. 91 m²
  3. 96 m²
  4. 99 m²
  5. 125 m²

Question 11

A palindrome reads the same forwards and backwards, for example 121 or 444. What is the digit sum of the largest three-digit palindrome divisible by 6?
  1. 16
  2. 18
  3. 20
  4. 21
  5. 24

Question 12

A rectangle is split into three equal-area pieces: an equilateral triangle of side 4 cm and two trapezoids, as shown. Find the shorter parallel side of a trapezoid.

Question illustration
  1. $\sqrt2$
  2. $\sqrt3$
  3. $2\sqrt2$
  4. 3
  5. $2\sqrt3$

Question 13

Fill the 2 × 4 table with A, B, C, D so that each letter occurs exactly once in each row and in each of the three 2 × 2 squares. How many fillings are possible?

Question illustration
  1. 12
  2. 24
  3. 48
  4. 96
  5. 198

Question 14

Three coloured disks overlap as in Figure 1, then touch pairwise as in Figure 2. In Figure 1 the visible black area is seven times the white disk’s area. What is the ratio of visible black areas in Figures 1 and 2?

Question illustration
  1. 3:1
  2. 4:3
  3. 6:5
  4. 7:6
  5. 9:7

Question 15

Mary’s granddaughter is born today. In two years the ages of Mary, her daughter and her granddaughter will all be even and their product will be 2024. How old is Mary today?
  1. 42
  2. 44
  3. 46
  4. 48
  5. 50

Question 16

P is inside equilateral triangle ABC. The three indicated segments through P, parallel to its sides, have lengths 2 m, 3 m and 6 m. What is the perimeter of ABC?

Question illustration
  1. 22 m
  2. 26 m
  3. 33 m
  4. 39 m
  5. 44 m

Question 17

Each circle contains a number. Each square shows the product at its four vertices. Find the product in the eight bold circles.

Question illustration
  1. 20
  2. 40
  3. 80
  4. 120
  5. 480

Question 18

An n×n×n cube is built from unit cubes and painted on its surface. The number of small cubes with exactly one painted face equals the number with no painted faces. Find n.
  1. 4
  2. 6
  3. 7
  4. 8
  5. 10

Question 19

The illustrated net consists of unit squares and equilateral triangles of side 1. After folding into the shown solid, what is the distance AB?

Question illustration
  1. $1+\sqrt2$
  2. $1+\sqrt3$
  3. $2\sqrt2$
  4. $\sqrt5$
  5. $5/2$

Question 20

Vlado ran 31 races over five years, with a strictly increasing positive number each year. In year five he ran three times as many as in year one. How many did he run in year four?
  1. 6
  2. 7
  3. 8
  4. 9
  5. 10