IKMC 2024 - Grades 11-12 - Set 2 - Self-study

Question 1

A pattern uses identical irregular pentagonal tiles. Which tile fills the hole so that a self-intersecting loop is formed?

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Question 2

Which number is two less than a multiple of 10, two more than a square, and twice a prime?

Question 3

A kangaroo cuts a pizza into six equal sectors, eats one, and rearranges the five remaining sectors around the centre with equal gaps. What is the angle of each gap?

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5°
8°
9°
10°
12°

Question 4

Juuso draws coordinate axes with positive x pointing left and positive y pointing down. Which graph represents y=x+1 in his coordinate system?

Question 5

Kaito has manipulated a die. Each of 2, 3, 4 and 5 still has probability 1/6, but rolling a 6 is twice as likely as rolling a 1. What is the probability of rolling a 6?

$1/4$
$1/6$
$7/36$
$2/9$
$5/18$

Question 6

Which expression equals $16^{15}+16^{15}+16^{15}+16^{15}$?

$16^{19}$
$4^{31}$
$4^{60}$
$16^{60}$
$4^{122}$

Question 7

Colour all squares and triangles of the pattern so cells that share an edge or even just a corner always have different colours. What is the minimum number of colours?

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Question 8

Six glasses stand on a table, all with their open ends up. In each move exactly four glasses are turned over. What is the least number of moves needed to have all six upside down?

Question 9

Starting from 1, a student repeatedly multiplies by either 6 or 10. Which number can he not obtain?

$2^{100}3^{20}5^{80}$
$2^{90}3^{20}5^{80}$
$2^{90}3^{20}5^{70}$
$2^{110}3^{80}5^{30}$
$2^{50}5^{50}$

Question 10

A black path and a grey path each divide a park into two equal areas. Which relation among the three marked intervening regions A, B, C must hold?

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$A=C$
$B=A+C$
$B=\frac12(A+C)$
$B=\frac23(A+C)$
$B=\frac35(A+C)$

Question 11

Exactly one of these statements about a positive integer n is true. Which one?

n is divisible by 3
n is divisible by 6
n is odd
n=2
n is prime

Question 12

A tetrahedron has edge lengths 5,6,7,8,9,10. M, N, P, Q, R, S are the edge midpoints shown. Find the total length of the closed polyline MNPQRSM.

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Question 13

Quadrilateral ABCD has right angles at B and C, with AB=4, BC=8, CD=2. What is the smallest possible AX+DX for X on segment BC?

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$9\sqrt2$
12
13
10
None of the previous choices

Question 14

John builds a 3 × 3 × 3 cube from 27 black or white unit cubes. Exactly half the outer surface must be black. What is the minimum number of black unit cubes?

14
13
12
11
None of the previous choices

Question 15

Inside a square of side 6 cm, a diagonal, a semicircle and a quarter-circle are drawn as shown. Find the total shaded area in cm².

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$9$
$3\pi$
$6\pi-9$
$10\pi/3$
$12$

Four squares are arranged as shown above a horizontal baseline. The smaller side lengths are a, b, c. Vertices A, C are opposite vertices of the large square, and B lies on one of its sides. Which expression equals its side length?

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$\frac12(a+b+c)$
$\sqrt{a^2+b^2+c^2}$
$\sqrt{(a+b)^2+c^2}$
$\sqrt{(b-a)^2+c^2}$
$\sqrt{a^2+ab+b^2+c^2}$

Question 17

Given two positive numbers $p<q$, which expression is largest?

$\frac{p+3q}{4}$
$\frac{p+2q}{3}$
$\frac{p+q}{2}$
$\frac{2p+q}{3}$
$\frac{3p+q}{4}$

$40/11$
$45/12$
$63/20$
$3$
$47/14$