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IKMC 2019 - Grades 9-10 - Self-study

Question 1

Two numbers have mean 2019. One is 2021. What is the other?
  1. 2017
  2. 2018
  3. 2020
  4. 2022
  5. 2023

Question 2

Which expression has the largest value?
  1. $2^3/3^2$
  2. $3^2/2^3$
  3. $3^2-2^3$
  4. $2^3-3^2$
  5. $(3-2)(2-3)$

Question 3

A barber writes SHAVE on a board so a client sees the word correctly in a mirror. How should the board be written?
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Question 4

How many distinct sums can appear when three standard six-sided dice are rolled together?
  1. 14
  2. 15
  3. 16
  4. 17
  5. 18

Question 5

Five identical marked glasses contain water. Four have equal amounts; which contains a different amount?
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Question 6

A park has five gates. Monica enters through one and exits through a different gate. In how many ways can she do this?
  1. 25
  2. 20
  3. 16
  4. 15
  5. 10

Question 7

Three kangaroos have distinct whole-number weights totalling 97 kg. What is the greatest possible weight of the lightest?
  1. 1 kg
  2. 30 kg
  3. 31 kg
  4. 32 kg
  5. 33 kg

Question 8

Nine identical squares form the shown grid. Which relation between the marked angles is true?

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  1. $\alpha=\beta$
  2. $2\alpha+\beta=90^\circ$
  3. $\alpha+\beta=60^\circ$
  4. $2\beta+\alpha=90^\circ$
  5. $\alpha+\beta=45^\circ$

Question 9

Each picture is a unit square with some shading. Which has the greatest total shaded area?
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Question 10

Three five-digit numbers are written on overlapping paper strips as shown; three digits are hidden. Their sum is 57263. Which are the hidden digits?

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  1. 0,2,2
  2. 1,2,9
  3. 2,4,9
  4. 2,7,8
  5. 5,7,8

Question 11

A square has vertices A, B, C, D labelled clockwise. An equilateral triangle has vertices A, E, C also labelled clockwise. What is angle CBE in degrees?
  1. 30
  2. 45
  3. 135
  4. 145
  5. 150

Question 12

Distinct positive integers a, b, c, d are chosen from 1 to 10. What is the least possible value of $a/b+c/d$?
  1. $2/10$
  2. $3/19$
  3. $14/45$
  4. $29/90$
  5. $25/72$

Question 13

The rectangular flag of Kanguria has side lengths in ratio 3:5 and is divided into four equal-area rectangles as shown. What is the ratio of the side lengths of the white rectangle?

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  1. 1:3
  2. 1:4
  3. 2:7
  4. 3:10
  5. 4:15

Question 14

A 3 × 2 rectangle can be tiled by two L-shaped three-square pieces in the two ways illustrated. How many different tilings of the target shape by these same L-shaped pieces are possible? Rotations of pieces are allowed; count partitions of the fixed target, not labels on identical pieces. AI restored the missing rectangle border in the examples.

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  1. 1
  2. 2
  3. 3
  4. 4
  5. 48

Question 15

In a triathlon, biking is three quarters of the total distance and running one fifth. Swimming is 2 km. What is the total distance in kilometres?
  1. 10
  2. 20
  3. 38
  4. 40
  5. 60

Question 16

Triangle ABC is divided by DE, EF, CF into four triangles of equal area as shown. Determine AF: BD.

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  1. 1
  2. 9:8
  3. 8:7
  4. 7:6
  5. 6:5

Question 17

The pictured shape is made from three equal circles of radius R with collinear centres. The middle circle passes through the other two centres. What is the perimeter of the outer shape?

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  1. $10\pi R/3$
  2. $5\pi R/3$
  3. $2\pi R\sqrt3/3$
  4. $2\pi R\sqrt3$
  5. $4\pi R$

Question 18

The digits of the seven-digit telephone number $\overline{aaabbbb}$ sum to the two-digit number $\overline{ab}$. Find a+b.
  1. 8
  2. 9
  3. 10
  4. 11
  5. 12

Question 19

Sixty apples and sixty pears are all packed into boxes. Each box has the same number of apples, and no two boxes have the same number of pears. What is the largest possible number of boxes?
  1. 20
  2. 15
  3. 12
  4. 10
  5. 6

Question 20

The diagram is a net of an octahedron. When folded, which labelled edge coincides with the edge marked x?

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  5. 5