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IKMC 2008 - Grades 7-8 - Self-study

Question 1

How many pieces of string are there in the picture?

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

Question 2

A class has 9 boys and 13 girls. Half the children have a cold. What is the smallest possible number of girls with a cold?
  1. 0
  2. 1
  3. 2
  4. 3

Question 3

Six kangaroos eat six sacks of grass in six minutes. How many kangaroos eat 100 sacks in 100 minutes at the same rate?
  1. 100
  2. 60
  3. 6
  4. 600

Question 4

The four cells of a 2×2 table contain 2,3,4 and one unknown number. The first row sums to 9 and the second to 6. Find the unknown.

Question illustration
  1. 5
  2. 6
  3. 7
  4. 8

Question 5

The triangle and square have equal perimeters. The square side is 4 cm. What is the perimeter of the combined pentagon?

Question illustration
  1. 24 cm
  2. 28 cm
  3. 32 cm
  4. It depends on the triangle dimensions

Question 6

A florist has 24 white, 42 red and 36 yellow roses. What is the greatest number of identical bunches using every flower?
  1. 4
  2. 6
  3. 8
  4. 12

Question 7

All eight corners of a cube are cut off as shown. How many edges does the resulting solid have?

Question illustration
  1. 30
  2. 36
  3. 40
  4. Another answer

Question 8

Three straight lines meet at one point. Two angles shown are 108° and 124°. Find the grey angle.

Question illustration
  1. 52°
  2. 53°
  3. 54°
  4. 56°

Question 9

Ali has nine 2-cent coins; Saima has eight 5-cent coins. What is the least total number of coins they must exchange with each other so their amounts become equal?
  1. 4
  2. 5
  3. 12
  4. Impossible

Question 10

How many squares have all their vertices at the dots shown?

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

Question 11

Two buses on a circular route are spaced 25 minutes apart. How many extra buses are needed to reduce that interval by 60%?
  1. 2
  2. 3
  3. 5
  4. 6

Question 12

The printed problem describes “the French mathematician August de Morgan”, who said he was x years old in the year $x^2$, and says he died in 1899. When was he born?
  1. 1806
  2. 1848
  3. 1849
  4. Another answer

Question 13

Starting on the mainland, visit islands A,B,C,D by ferry. B connects only to A and the mainland; A and C connect to each other and to the mainland; D connects only to A. What is the minimum number of ferry rides needed to visit all four islands?
  1. 6
  2. 5
  3. 4
  4. 7

Question 14

Tom and Jerry each cut an identical rectangle into two rectangles. Tom’s two pieces each have perimeter 40 cm; Jerry’s each have perimeter 50 cm. What was the perimeter of each original rectangle?
  1. 40 cm
  2. 50 cm
  3. 60 cm
  4. 80 cm

Question 15

One face of a cube is cut along its two diagonals as shown. Which numbered nets are impossible?

Question illustration

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

Question 16

A,B,C,D lie on a straight line in some order. AB=13, BC=11, CD=14, DA=12. What is the distance between the two farthest points?
  1. 14
  2. 38
  3. 25
  4. Another answer

Question 17

Four tangent circles of radius 6 cm lie in a row inside the rectangle shown. P is the upper-right vertex, Q and R are the indicated tangency points. Find the area of triangle PQR.

Question illustration
  1. $27\text{ cm}^2$
  2. $45\text{ cm}^2$
  3. $54\text{ cm}^2$
  4. $108\text{ cm}^2$

Question 18

Seven cards numbered 1 to 7 lie in a box. The first sage takes three, the second takes two, and two remain. The first sage says, “I know the sum of your two cards is even.” What is the sum of the first sage’s cards?
  1. 10
  2. 12
  3. 9
  4. 15

Question 19

In isosceles triangle ABC with base BC, the internal bisector CD of angle C has length equal to BC. Find angle CDA.
  1. 100°
  2. 108°
  3. 120°
  4. Impossible to determine

Question 20

An 11×11×11 wooden cube is made of 11³ unit cubes. What is the greatest number of unit cubes visible from one viewpoint?
  1. 329
  2. 330
  3. 331
  4. 332