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IKMC 2025 - Grades 11-12 - Self-study

Question 1

The year 2025 is a perfect square: $2025=45^2$. How many years pass until the next square-numbered year?
  1. 25
  2. 91
  3. 121
  4. 500
  5. 2025

Question 2

Mike divides $\sqrt{11}$ by 3 to obtain x. Where is x on the number line?
  1. Between 0 and 1
  2. Between 1 and 2
  3. Between 2 and 3
  4. Between 3 and 4
  5. Between 4 and 5

Question 3

Vasily has 20 balls, each yellow, green, blue or black. Exactly 17 are not green, 15 are not black, and 12 are not yellow. How many are blue?
  1. 8
  2. 7
  3. 6
  4. 4
  5. 3

Question 4

Which interval contains $77\times777$?
  1. Between 7 and 77
  2. Between 77 and 777
  3. Between 777 and 7,777
  4. Between 7,777 and 77,777
  5. Between 77,777 and 777,777

Question 5

Which expression equals $\sqrt{16^{16}}$?
  1. $4^4$
  2. $4^8$
  3. $4^{16}$
  4. $8^8$
  5. $16^4$

Question 6

The pictures are the first three in a sequence. How many dots are in the fifth picture?

Question illustration
  1. 72
  2. 74
  3. 76
  4. 78
  5. 80

Question 7

Five stones strike a window at A, B, C, D and E. New cracks end at the window edge or at an existing crack. In which order did the impacts occur?

Question illustration
  1. DACBE
  2. ABCDE
  3. BDACE
  4. BCDAE
  5. DCABE

Question 8

A packet used to contain five chocolate bars and now contains four for the same price. By what percentage has the price of each bar increased?
  1. 10%
  2. 20%
  3. 25%
  4. 30%
  5. 50%

Question 9

ABCD is a rectangle. Quadrilateral ABPD has area $4\text{ cm}^2$ and $DP=CD/3$. What is the area of ABCD?

Question illustration
  1. $5\text{ cm}^2$
  2. $5.5\text{ cm}^2$
  3. $6\text{ cm}^2$
  4. $6.5\text{ cm}^2$
  5. $7\text{ cm}^2$

Question 10

In the unit square $0\le x,y\le1$, a point is black exactly when the first digit after the decimal point of both x and y is odd. Which picture results?
  1. Question illustration
  2. Question illustration
  3. Question illustration
  4. Question illustration
  5. Question illustration

Question 11

Robert wants four points whose six pairwise distances are all different. Which point must he remove?

Question illustration
  1. A
  2. B
  3. C
  4. D
  5. E

Question 12

Among ten distinct positive integers, exactly five are divisible by 5 and exactly seven by 7. If M is their largest, what is the smallest possible M?
  1. 105
  2. 77
  3. 75
  4. 63
  5. A different value

Question 13

A quarter-circle SP has centre O and radius r. The two grey regions between it and triangle ORP have equal areas. What is OR?

Question illustration
  1. $\pi r/2$
  2. $3r/2$
  3. $\pi r$
  4. $2/\pi$
  5. $\pi/(2r)$

Question 14

The graphs of $y=ax+b$ and $y=cx+d$ are as shown. What is certain about $ab+cd-(ac+bd)$?

Question illustration
  1. It is negative
  2. It is not positive
  3. It is positive
  4. It equals zero
  5. None of these is certain

Question 15

A ball of wool initially has diameter 30 cm. After Grandma knits 70 socks, its diameter is 15 cm. How many more socks can she knit?
  1. 70
  2. 50
  3. 30
  4. 20
  5. 10

Question 16

One kangaroo stands in each cell of a 4 × 4 chessboard. Every move, each jumps to an orthogonally adjacent cell, staying on the board; sharing a cell is allowed. What is the maximum number of empty cells after 100 moves?

Question illustration
  1. 15
  2. 14
  3. 12
  4. 10
  5. 8

Question 17

The five-digit number $\overline{N18NN}$ is divisible by 18. Which statement about digit N is true?
  1. Exactly one such N
  2. Exactly two such N
  3. Exactly three such N
  4. No such N
  5. More than three such N

Question 18

The black semicircle has area $12\text{ cm}^2$. What is the area of the large quarter-circle?

Question illustration
  1. $42\text{ cm}^2$
  2. $36\text{ cm}^2$
  3. $32\text{ cm}^2$
  4. $30\text{ cm}^2$
  5. $25\text{ cm}^2$

Question 19

Three square Martians and three round Jupiterians sit as shown. One has the spaceship key. One planet’s inhabitants always tell the truth and the other’s always lie. Each answers the question “Does either neighbour have the key? ” as shown. Who has it?

Question illustration
  1. A
  2. B
  3. C
  4. D
  5. E

Question 20

B and C lie on the diameter AD of a semicircle, and E, F, G, H lie on its arc as shown. How many triangles have vertices among these eight points?

Question illustration
  1. 15
  2. 50
  3. 51
  4. 52
  5. 54