IKMC 2022 - Grades 11-12 - Self-study

Question 1

Martin’s phone shows last week’s use of four apps, sorted by decreasing time. This week he uses two apps for half as long and the other two for the same time as last week. Which diagram cannot show this week’s use?

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Question 2

How many positive three-digit integers are divisible by 13?

Question 3

Bella is older than Charly and younger than Lily. Teddy is older than Bella. Which pair can be the same age?

Charly and Teddy
Teddy and Lily
Lily and Charly
Bella and Lily
Teddy and Bella

Question 4

Which number is not divisible by its own digit sum?

2022
2023
2024
2025
2027

Question 5

$5^8$ pencils are distributed equally among 25 empty boxes. How many pencils go in each box?

$25^3$
$25^2$
$5^3$
$5^2$
$5$

Question 6

The product of the digits of a ten-digit number is 15. What is the sum of its digits?

Question 7

Four circles of radius 1 intersect as shown. What is the perimeter of the grey region?

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$\pi$
$3\pi/2$
A number between $3\pi/2$ and $2\pi$
$2\pi$
$\pi^2$

Question 8

List in increasing order all integers from 2 to 2022 that use only digits 0 and 2. Which is the middle number?

200
220
222
2000
2002

Question 9

How many real solutions does $(x-2)^2+(x+2)^2=0$ have?

Question 10

A, B, C, D lie on a line in this order. AC=12 cm and BD=18 cm. What is the distance between the midpoints of AB and CD?

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6 cm
9 cm
12 cm
13 cm
15 cm

Question 11

Four lines through one point create eight equal angles. Which labelled black arc has the same length as the circumference of the small grey circle?

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Question 12

Nonzero real numbers a, b, c make $-2a^4b^3c^2$ and $3a^3b^5c^{-4}$ have the same sign. Which statement must hold?

$ab>0$
$b<0$
$c>0$
$bc>0$
$a<0$

Question 13

The water meter shows 91.876 m³, with all five displayed digits different. What is the least further water use before all displayed digits are different again?

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0.006 m³
0.034 m³
0.086 m³
0.137 m³
1.048 m³

Question 14

A square is divided into two squares and two rectangles. The shaded quadrilateral’s vertices are midpoints of sides of the smaller squares, and its area is 3. What is the non-shaded area of the large square?

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Question 15

Find the greatest common divisor of $2^{2021}+2^{2022}$ and $3^{2021}+3^{2022}$.

$2^{2021}$
$1$
$2$
$6$
$12$

Question 16

Sixteen towns are connected by the roads shown. A power plant supplies its own town and every town joined to it by one road. What is the minimum number of plants needed to supply all towns?

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Question 17

Eight players are randomly placed in a knockout tournament: four first-round games, two second-round games, and a final. Anita and Martina beat everyone else, and Anita beats Martina. What is the probability that Martina reaches the final?

$1$
$1/2$
$2/7$
$3/7$
$4/7$

Question 18

A cuboid with surface area X is cut by six planes parallel to its faces, two in each direction, into 27 solids. What is the total surface area of all the pieces?

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X
2 X
3 X
4 X
Depends on the positions of the planes

Question 19

Five numbers have arithmetic mean 24. The three smallest have mean 19, and the three largest mean 28. What is the median?

Question 20

A circle centred at (0,0) has radius 5. How many points on its circumference have both coordinates integers?

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