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IKMC 2023 - Grades 11-12 - Self-study

Question 1

Simplify $\dfrac{7777^2}{5555\cdot2222}$.
  1. $1$
  2. $7/10$
  3. $49/10$
  4. $77/110$
  5. $49$

Question 2

Julia rolls five ordinary six-sided dice. Their sum is 19. What is the largest possible number of sixes?
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Question 3

A cylindrical tin is 15 cm high and its base circumference is 30 cm. An ant follows the bold route from A at the bottom to B at the top, using upward vertical segments and horizontal circular arcs. Front portions are solid and back portions dashed. What total distance does it travel?

Question illustration
  1. 45 cm
  2. 55 cm
  3. 60 cm
  4. 65 cm
  5. 75 cm

Question 4

A is a 2023-digit number whose every digit is 1. What is the sum of the digits of 1111 A?
  1. 8080
  2. 8083
  3. 8086
  4. 8092
  5. 8101

Question 5

Emma colours the three strips of the flag, using one of four colours for each strip. Adjacent strips must have different colours. How many colourings are possible?

Question illustration
  1. 24
  2. 27
  3. 32
  4. 36
  5. 64

Question 6

A positive integer n is called twoprime if its only three distinct positive divisors are 1,2, and n. How many twoprime integers exist?
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Question 7

What is the units digit of $(5^5+1)(5^{10}+1)(5^{15}+1)$?
  1. 0
  2. 1
  3. 3
  4. 5
  5. 6

Question 8

Evaluate $2^{0^{2^3}}+0^{2^{3^2}}+2^{3^{2^0}}+3^{2^{0^2}}$.
  1. 3
  2. 4
  3. 7
  4. 12
  5. More than 100

Question 9

A row contains 23 animals, each a beaver or kangaroo. Every animal has at least one kangaroo immediately next to it. What is the largest possible number of beavers?
  1. 7
  2. 8
  3. 10
  4. 11
  5. 12

Question 10

A square of area 84 is split into four equal squares. The upper-left square is blackened, and the lower-right square is repeatedly subdivided in the same way, infinitely often. What is the total black area?

Question illustration
  1. 24
  2. 28
  3. 31
  4. 35
  5. 42

Question 11

Place 1 through 9 once each in the nine cells shown, with 7 and 9 already fixed. The sum in any three consecutive cells must be divisible by 3. How many fillings are possible?

Question illustration
  1. 9
  2. 12
  3. 15
  4. 18
  5. 24

Question 12

Two equilateral triangles of different sizes overlap in a hexagon with opposite sides parallel. Four side lengths are shown. Find the hexagon’s perimeter.

Question illustration
  1. 64
  2. 66
  3. 68
  4. 70
  5. 72

Question 13

Five numbers $a_1,a_2,a_3,a_4,a_5$ have sum S and satisfy $a_k=k+S$ for $1\le k\le5$. Find S.
  1. $15/4$
  2. $-15/4$
  3. $-15$
  4. 15
  5. Another number

Question 14

All six edge lengths of the tetrahedron are integers. Four are shown. What is the sum of the other two?

Question illustration
  1. 9
  2. 10
  3. 11
  4. 12
  5. 13

Question 15

How many integer pairs $(m,n)$ satisfy $|2m-2023|+|2n-m|\le1$?
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Question 16

Write $5^{5^6}$ as $n^n$ for a natural number n. What is n?
  1. $5\cdot5^4$
  2. $5\cdot5^5$
  3. $5^{30}$
  4. 30
  5. $25^{25}$

Question 17

Leon draws a closed path on a cuboid’s surface. Which net can represent his path?
  1. Question illustration
  2. Question illustration
  3. Question illustration
  4. Question illustration
  5. Question illustration

Question 18

For a positive integer n, $n!=1\cdot2\cdots n$. Given $N!=6!\cdot7!$, what is the sum of the digits of N?
  1. 1
  2. 2
  3. 4
  4. 8
  5. 9

Question 19

All graphs $y=x^3+3x^2+ax+2a+4$ pass through a common point independent of a. What is the sum of its coordinates?
  1. 2
  2. 4
  3. 7
  4. 8
  5. Another number

Question 20

A pentagon is subdivided as shown. The numbers are triangle areas. Find the area P of the grey quadrilateral.

Question illustration
  1. 15
  2. $31/2$
  3. 16
  4. 17
  5. Another number