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IKMC 2023 - Grades 9-10 - Set 2 - Self-study

Question 1

A grey circle with two holes is placed on a clock face as shown. It is rotated about the centre until 10 appears in one hole. Which numbers could appear in the other hole?

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

Question 2

Maria ran to catch the subway, got off two stops later, and then walked to school. Which speed–time graph best represents her journey?
  1. Question illustration
  2. Question illustration
  3. Question illustration
  4. Question illustration
  5. Question illustration

Question 3

The positive integers m and n are both odd. Which expression is also odd?
  1. m (n+1)
  2. (m+1)(n+1)
  3. m+n+2
  4. mn+2
  5. m+n

Question 4

A square of side 10 cm contains a smaller square of side 4 cm as shown. Corresponding sides are parallel. What percentage of the large square is shaded?

Question illustration
  1. 25%
  2. 30%
  3. 40%
  4. 42%
  5. 45%

Question 5

Today is Thursday. What day will it be in 2023 days?
  1. Tuesday
  2. Wednesday
  3. Thursday
  4. Friday
  5. Saturday

Question 6

The rectangle is divided into 30 equal squares as shown. The shaded region has perimeter 240 cm. What is the rectangle’s area?

Question illustration
  1. 480 cm²
  2. 750 cm²
  3. 1080 cm²
  4. 1920 cm²
  5. 2430 cm²

Question 7

The ages of a family of five total 80. The two youngest are 6 and 8. What was the sum of the family’s ages seven years ago?
  1. 35
  2. 36
  3. 45
  4. 46
  5. 66

Question 8

A wooden fence has vertical planks, with four horizontal planks joining each vertical plank to the next. The first and last planks are vertical. Which total number of planks is possible?
  1. 95
  2. 96
  3. 97
  4. 98
  5. 99

Question 9

In how many ways can positive integers a and b satisfy a/5=7/b?
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Question 10

After 200 chess games, my winning rate is exactly 49%. What is the smallest number of additional games needed to increase it to exactly 50%?
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Question 11

Jenni reduces her shower time by one quarter and the water flow rate by one quarter. By what fraction does she reduce the total water used?
  1. $1/4$
  2. $3/8$
  3. $5/8$
  4. $5/12$
  5. $7/16$

Question 12

Three squares have side lengths 3 cm, 5 cm and 8 cm as shown. What is the shaded trapezium’s area in cm²?

Question illustration
  1. 13
  2. $55/4$
  3. $61/4$
  4. $65/4$
  5. $69/4$

Question 13

A 95 m wire is cut into three pieces, each 50% longer than the previous piece. How long is the largest piece?
  1. 36 m
  2. 42 m
  3. 45 m
  4. 46 m
  5. 48 m

Question 14

M and N are midpoints of two sides of the rectangle. What fraction of its area is shaded?

Question illustration
  1. $1/6$
  2. $1/5$
  3. $1/4$
  4. $1/3$
  5. $1/2$

Question 15

Pentagon ABCDE is split into four triangles of equal perimeter. ABC is equilateral; AEF, DFE and CDF are congruent isosceles triangles. What is the ratio of the pentagon’s perimeter to the perimeter of ABC?

Question illustration
  1. $2/1$
  2. $3/2$
  3. $4/3$
  4. $5/3$
  5. $5/2$

Question 16

A tower has blocks numbered 1 to 90, as shown. Bob takes the top three blocks at a time, preserving their order, to build a new tower. After all blocks are moved, how many lie between blocks 39 and 40?

Question illustration
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Question 17

Every third step of a staircase of 2023 steps is black, beginning as shown. Anita climbs one step at a time, alternating feet, starting with either foot. What is the smallest number of black steps she steps on with her right foot?

Question illustration
  1. 0
  2. 333
  3. 336
  4. 337
  5. 674

Question 18

A two-digit number is power-less if neither digit can be written as an integer raised to a power greater than 1. For example 53 qualifies, but 54 does not because 4=2². Which is a common divisor of the smallest and largest power-less numbers?
  1. 3
  2. 5
  3. 7
  4. 11
  5. 13

Question 19

A square of side 30 cm is divided into nine equal smaller squares. It contains circles of radii 5 cm (bottom right), 4 cm (top left), and 3 cm (top right), as shown. What is the shaded area?

Question illustration
  1. 400 cm²
  2. 500 cm²
  3. $(400+50\pi)\text{ cm}^2$
  4. $(500-25\pi)\text{ cm}^2$
  5. $(500+25\pi)\text{ cm}^2$

Question 20

Tim computes the mean of five different prime numbers and obtains an integer. What is the smallest integer possible?
  1. 2
  2. 5
  3. 6
  4. 12
  5. 30