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IKMC 2023 - Grades 11-12 - Set 2 - 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

Giulia 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. The thicker path is black at the front and grey at the back. 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

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 5

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 6

How many pairs of positive integers $x,y$ satisfy $x+2y=2^{10}$?
  1. $2^9-1$
  2. $2^9$
  3. $2^9+1$
  4. $2^9+2$
  5. 0

Question 7

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 8

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 9

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 10

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

Question 11

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 12

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 13

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. None of the previous choices

Question 14

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. None of the previous choices

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

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 17

Write $5^{5^6}$ as $n^n$ for a natural number n. What is n?
  1. $5^{30}$
  2. $5^6$
  3. $5^5$
  4. 30
  5. 11

Question 18

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 19

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. 18

Question 20

How many integers are factors of $2^{20}3^{23}$ but are not factors of $2^{10}3^{20}$?

Editorial convention: the supplied answer key uses positive factors, the usual contest convention; scoring here follows that convention. The signed-integer interpretation is discussed in the solution.
  1. 13
  2. 30
  3. 273
  4. 460
  5. None of the previous choices