IKMC 2022 - Grades 11-12 - Set 2 - Self-study

Question 1

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

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

Question 2

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

Question 3

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

Charlie and Teddy
Teddy and Lily
Lily and Charlie
Bella and Lily
Teddy and Bella

Question 4

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

Question 5

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

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

Question 6

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 7

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

Question 8

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 9

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 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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15 cm
12 cm
18 cm
6 cm
9 cm

Question 11

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

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 13

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

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

Question 14

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 15

Which pair of building blocks can be joined, using rigid rotations, to form the shown solid?

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Question 16

Martina plays in an eight-player knockout tournament. She beats everyone except Ash, who beats everybody. Players are randomly assigned to four first-round pairs; winners proceed to two second-round matches, then the final. What is the probability that Martina does not reach the final?

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

Question 17

A cuboid of surface area $S$ is cut by six planes as shown, two parallel to each pair of opposite faces. Their distances from the faces are arbitrary, giving 27 smaller cuboids. What is their total surface area in terms of $S$?

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$2S$
$5S/2$
$3S$
$4S$
None of the previous choices

Question 18

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

Question 19

Two rectangles of dimensions 1 × 5 and 2 × 3 are inscribed in a triangle as shown. What is the triangle’s height from A?

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3
$7/2$
$8/3$
$16/5$
None of the previous choices

Question 20

A rectangle is divided into 11 smaller rectangles, all similar to the original. The smallest rectangles have the same orientation as the original, and their lower sides have length 1. Find the large rectangle’s perimeter.

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