WA Junior Mathematics Olympiad 2002

Individual Questions 100 minutes (one hour and 40 minutes)

General instructions: No working need be given for Questions 1 to 9. Calculators are not permitted. For Questions 1 to 9, write the answer in the answer grid. Write your answer to Question 10 in the space provided.

1. Simplify . [1 mark]

2. Four different positive integer numbers a, b, c, d satisfy the following relations:

+ + = 1 , + + = 1 , + + = 1Find d. [2 marks]

3. Three athletes Ahmad, Bill and Claire are preparing to take part in a high jump competition. At the same time some of the spectators are discussing their chances:spectator X : ‘I think Ahmad will be first’,spectator Y: ‘I am sure that Claire will not be the last’,spectator Z : ‘Bill will not take first place’.After the competition it turned out that only one of the spectators was right, while the other two were wrong. Where did Claire finish?

[2 marks]

4. In a rectangle ABCD, O is the intersection point of the diagonals AC and BD and BD = 10 cm. Find the length of BC if it is known that the point D lies on the perpendicular bisector of the segment AO.

[2 marks]

5. Three positive numbers are given such that :(i) the first of the numbers is half the second;(ii) the product of the first and the second number is equal to the sum of

the second and the third number;(iii) the third number is three times as large as the second.Find the first of the given numbers. [2 marks]

6. Find the smallest positive integer divisible by 15 whose every digit is 0 or 1. [2 marks]

7. A rectangle ABCD has sides AB = CD = 34 cm. E is a point on CD such that CE = 9cm, ED = 25 cm, and AEB = 900. What is the length of BC? [3 marks]

8. I have 6 cats, two white, 2 black and 2 orange, with a male and female of each colour. I want to put them in a row of 6 boxes. The orange cats are friends and have to be put side by side, but the black cats fight and must not be side by side. In how many ways can I arrange the cats?

[3 marks]

9. Five different integer numbers a, b, c, d, e (not necessarily positive) are such that

(4 − a)(4 − b)(4 − c)(4 − d)(4 − e) = 12 .Find the sum a + b + c + d + e. [4 marks]

10. All faces of a cube are divided into four equal squares and each small square is painted red, blue or green in such a way that any two small squares that have a common side are painted in different colours. Is it true that the numbers of red, blue and green squares must be the same? Give reasons for your answer. [4 marks]

Western AustralianJunior Mathematics Olympiad 2002

Team Questions 45 minutes

1. You are given 9 coins of the same denomination, and you know that one of them is counterfeit and that it is lighter than the others. You have a pan balance which means you can put any number of coins on each side and the balance will tell you which side is heavier, but not how much heavier. Explain how you can find the counterfeit coin in exactly two weighings.

[4 marks]

2. If you are given 25 coins of the same denomination, and you know that one of them is counterfeit and that it is lighter than the others, explain how to find the counterfeit coin by using at most 3 weighings on the pan balance.

[8 marks]

3. It is known that there is one counterfeit coin in a collection of 70 and that it is lighter than the others. What is the least number of weight trials on a pan balance necessary to identify the counterfeit coin? Explain how you obtained your answer. [12 marks]

4. It is known that there is one counterfeit coin in a collection of 9 and it is known that its weight is different from that of a genuine coin, however it is not known whether the counterfeit coin is lighter or heavier than a genuine one. Show that by using at most 3 weighings in the pan balance you can identify the counterfeit coin and determine whether it is lighter or heavier than a genuine coin. [16 marks]