IKMC 2024 - Grades 11-12 - Self-study

Question 1

The angles of a triangle are in the ratio 1:3:5. What is its largest angle?

80°
90°
100°
110°
120°

Question 2

A pattern uses identical irregular pentagonal tiles. Which tile fills the hole so that a closed curve is formed?

Question illustration

Question 3

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

Question 4

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?

Question illustration

5°
8°
9°
10°
12°

Question 5

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

Question 6

A manipulated die still rolls each of 2,3,4,5 with probability 1/6, but a 6 is twice as likely as a 1. What is the probability of 6?

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

Question 7

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

$4^{31}$
$4^{38}$
$4^{60}$
$4^{120}$
$4^{122}$

Question 8

Six coins initially show heads. Each move flips exactly four coins. What is the smallest number of moves that can leave all six tails up?

Question 9

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

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

Question 10

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

Question illustration

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

Question 11

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
Another number

Question 12

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

Question illustration

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

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?

Question illustration

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

Question 14

Given $0<x<y$, which expression is largest?

$\frac{x+3y}{4}$
$\frac{x+2y}{3}$
$\frac{x+y}{2}$
$\frac{2x+y}{3}$
$\frac{3x+y}{4}$

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

Question 20

Two candles of equal initial length are lit together. One burns completely in 4 hours, the other in 5 hours, at constant rates. After how many hours is one exactly three times as long as the other?

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