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IKMC 2009 - Grades 7-8 - Self-study

Question 1

Which of the following is an even number?
  1. $2009$
  2. $2+0+0+9$
  3. $200-9$
  4. $200\times9$
  5. $200+9$

Question 2

At a party there were 4 boys and 4 girls. Boys only danced with girls and girls only danced with boys. At the end of the evening each person was asked how many people they had danced with. The boys gave the answers 3, 1, 2, 2 and three of the girls answered 2. Which answer did the fourth girl give?
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Question 3

The star shown in the picture is made by fitting together 12 congruent equilateral triangles. The perimeter of the star is 36 cm. What is the perimeter of the grey hexagon?

Question illustration
  1. 6 cm
  2. 12 cm
  3. 18 cm
  4. 24 cm
  5. 30 cm

Question 4

Harry delivers newspapers in Long street. He must deliver a paper to every house with an odd house number. If the first house is number 15 and the last is number 53, to how many houses does Harry deliver?
  1. 19
  2. 20
  3. 27
  4. 38
  5. 53

Question 5

In the picture the large square has an area of 1. What is the area of the small black square?

Question illustration
  1. $\frac1{100}$
  2. $\frac1{300}$
  3. $\frac1{600}$
  4. $\frac1{900}$
  5. $\frac1{1000}$

Question 6

The product of four different natural numbers is 100. What is the sum of the four numbers?
  1. 10
  2. 12
  3. 15
  4. 18
  5. 20

Question 7

In a park there are some cats and dogs. The number of cats’ feet is double the number of dogs’ noses. The number of cats is …… of the number of dogs.
  1. double the size
  2. half the size
  3. the same size
  4. a quarter the size
  5. a sixth of the size

Question 8

In the diagram QSR is a straight line. $\angle QPS=12^\circ$ and $PQ=PS=RS$. How big is $\angle QPR$?

Question illustration
  1. $36^\circ$
  2. $42^\circ$
  3. $54^\circ$
  4. $60^\circ$
  5. $84^\circ$

Question 9

A lift can carry either 12 adults or 20 children. What is the maximum number of children that could travel in the lift with 9 adults?
  1. 3
  2. 4
  3. 5
  4. 6
  5. 8

Question 10

Which of the following is made using more than one piece of string?

Question illustration
  1. I, III, IV and V
  2. I, III and V
  3. III, IV and V
  4. all
  5. None of these answers

Question 11

For how many positive whole numbers a do $a^2$ and $a^3$ have the same number of digits?
  1. 0
  2. 3
  3. 4
  4. 9
  5. infinitely many

Question 12

What is the minimum number of dots that must be taken away from the picture so that no three of the remaining dots lie on a straight line?

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

Question 13

Nick measured all 6 angles in two triangles. One triangle was acute angled and the other obtuse angled. He noted four angles: 120°, 80°, 55° and 10°. What is the smallest angle in the acute triangle?
  1. 45°
  2. 10°
  3. 5°
  4. 55°
  5. not possible to answer

Question 14

What fraction of the largest square is grey?

Question illustration
  1. $\frac14$
  2. $\frac15$
  3. $\frac25$
  4. $\frac38$
  5. $\frac13$

Question 15

On the island of the truth tellers and the liars, 25 people stand in a line. The person at the front claims that everybody behind him is a liar. Everybody else claims that the person immediately in front of them is a liar. How many liars are in the line? (Truth tellers always tell the truth and liars always lie.)
  1. 0
  2. 12
  3. 13
  4. 24
  5. not possible to answer

Question 16

The diagram shows an object with 6 triangular faces. On each corner is a number (two are shown). The sum of the numbers on the corners of each face is the same. What is the sum of all 5 numbers?

Question illustration
  1. 9
  2. 12
  3. 17
  4. 18
  5. 24

Question 17

In the equation $\frac{E\times I\times G\times H\times T}{F\times O\times U\times R}=T\times W\times O$, each letter represents a certain digit (the same letter represents the same digit each time). How many different values can the expression $T\cdot H\cdot R\cdot E\cdot E$ have?

Editorial convention: the intended puzzle uses different digits for different letters. The source does not explicitly state this extra condition. The worked solution explains both readings.
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5

Question 18

We want to paint each square in the grid with colours P,Q,R,S, so that neighbouring squares always have different colours. Squares sharing a corner also count as neighbouring. Some squares are already painted. In which colour(s) could the grey square be painted?

Question illustration
  1. only Q
  2. only R
  3. only S
  4. either R or S
  5. it is not possible

Question 19

The diagram shows a regular nonagon. What is the size of the angle marked X?

Question illustration
  1. $40^\circ$
  2. $45^\circ$
  3. $50^\circ$
  4. $55^\circ$
  5. $60^\circ$

Question 20

A pattern is made out of white square tiles. The first three patterns are shown. How many tiles will be needed for the tenth pattern?

Question illustration
  1. 76
  2. 80
  3. 84
  4. 92
  5. 100