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IKMC 2013 - Grades 9-10 - Self-study

Question 1

The number 200013−2013 is not divisible by which number?
  1. 2
  2. 3
  3. 5
  4. 7
  5. 11

Question 2

Mary drew six identical squares, each containing a shaded region. How many shaded regions have perimeter equal to the perimeter of a square?

Question illustration
  1. 2
  2. 3
  3. 4
  4. 5
  5. 6

Question 3

Mrs Margareth buys 4 cobs of corn for each of her four family members. The sign says 1 cob costs 20 cents and every sixth cob is free. How much does she pay?

Question illustration
  1. 0.80 EUR
  2. 1.20 EUR
  3. 2.80 EUR
  4. 3.20 EUR
  5. 80 EUR

Question 4

Three numbers from 2,4,16,25,50,125 have product 1000. What is their sum?
  1. 70
  2. 77
  3. 131
  4. 143
  5. 145

Question 5

Six points are marked on a square grid with 1 × 1 cells. Kanga chooses three marked points as vertices of a triangle. What is the smallest possible area?

Question illustration
  1. $1/4$
  2. $1/3$
  3. $1/2$
  4. $1$
  5. $2$

Question 6

Which expression equals $4^{15}+8^{10}$?
  1. $2^{10}$
  2. $2^{15}$
  3. $2^{20}$
  4. $2^{30}$
  5. $2^{31}$

Question 7

The outside of a cube is painted grey and white so that it looks built from smaller grey and white cubes, as shown. Which net could give this painted cube?

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Question 8

n is the largest positive integer for which 4 n has three digits, and m is the smallest such positive integer. Find 4 n−4 m.
  1. 900
  2. 899
  3. 896
  4. 225
  5. 224

Question 9

The shown three-quarter circle with centre M and orientation arrow is rotated 90° anticlockwise about M, then reflected in the x-axis. Which picture is the final result?

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Question 10

Which value is largest?
  1. $\sqrt{20}\cdot\sqrt{13}$
  2. $\sqrt{20}\cdot13$
  3. $20\cdot\sqrt{13}$
  4. $\sqrt{201}\cdot3$
  5. $\sqrt{2013}$

Question 11

Triangle RZT is the image of equilateral triangle KZM under rotation about Z. In the diagram, angle MZR=70°. Find angle RKM.

Question illustration
  1. 20°
  2. 25°
  3. 30°
  4. 35°
  5. 40°

Question 12

The zigzag shown has six 1 cm by 1 cm squares and perimeter 14 cm. Continue the same zigzag to 2013 squares. What is its perimeter, in cm?

Question illustration
  1. 2022
  2. 4028
  3. 4032
  4. 6038
  5. 8050

Question 13

P, Q are opposite vertices of a regular hexagon, and R, S are midpoints of opposite edges, as shown. Its area is 60 cm². Find PQ×RS.

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  1. 40 cm²
  2. 50 cm²
  3. 60 cm²
  4. 80 cm²
  5. 100 cm²

Question 14

A class took a test. If every boy had scored 3 points more, the class average would be 1.2 points higher. What percentage of the class are girls?
  1. 20
  2. 30
  3. 40
  4. 60
  5. 50

Question 15

Rectangle ABCD is below the x-axis and left of the y-axis, with sides parallel to the axes. For each vertex, divide its y-coordinate by its x-coordinate. Which vertex gives the smallest value?

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  1. A
  2. B
  3. C
  4. D
  5. Depends on rectangle size

Question 16

On John’s birthday this year, he multiplied his age by his son’s age and obtained 2013. In which year was John born?
  1. 1981
  2. 1982
  3. 1953
  4. 1952
  5. More information is needed

Question 17

In triangle QPS, angle QPS=59° and PSQ=60°. In triangle QRS, angle RQS=61° and QRS=60°, as shown. Which listed segment is longest?

Question illustration
  1. PS
  2. RS
  3. QS
  4. QR
  5. PQ

Question 18

Ivana wants five consecutive integers such that three have the same sum as the other two. How many different sets can she write?

Editorial correction: the printed word “integers” is too broad for the choices. This package uses the intended positive integers; the literal unrestricted case is explained in the solution.
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Question 19

How many paths lead from A to B along the edges, travelling only in the indicated arrow directions?

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  1. 6
  2. 8
  3. 9
  4. 12
  5. 15

Question 20

Roo wants a six-digit number with even digit sum and odd digit product. Which statement about such a number is correct?
  1. Either two or four digits are even
  2. Such a number cannot exist
  3. An odd number of digits are odd
  4. All six digits can be different
  5. None of A–D is correct