# Here Are The Maths Questions Put To WA's Top 14-Year-Old Students

The University of Western Australia is gearing up for the state’s eighth annual WA Junior Mathematics Olympiad, in which year 8 and 9 students will compete for \$3500 in prizes.

Students are asked to solve a series of individual and team-based problems without a calculator, and the problems aren’t simple.

Committee director Greg Gamble said last year’s Olympiad was the toughest to date.

No one had a perfect score on the individual paper, although there were two students with 23 marks out of 25, and a few 22s.

Courtesy of the Western Australian Mathematical Olympiads Committee, here are last year’s individual questions. The answers are here, on the WAJO website.

1. A point X within a rectangle PQRS is such that XS = XR and the area of the triangle QXR is 7 square centimetres. How many square centimetres is the area of the rectangle? (1 mark)

2. A freeway is (in total) 16 metres wide, and streetlamps are placed in the middle of the road (along the median strip separating the two directions of traffic).

Each lamp lights a disc (i.e. the area within a circle) with diameter 20 metres. What is the maximum number of metres between the lamps in order that no part of the freeway is unlit? (1 mark)

3. In the picture of adjoining tiles below, tiles A to H are square. The area of tile F is 16 square units, the area of tiles B and G are each 25 square units.

How many square metres is the total area of tile D and tile H combined? (1 mark)

4. On Monday, the produce manager, Arthur Applegate, stacked the display case with 80 lettuces. By the end of the day, some of the lettuces had been sold.

On Tuesday, Arthur surveyed the display case and counted the lettuces that were left. He decided to add an equal number of lettuces. (He doubled the leftovers.) By the end of the day, he had sold the same number of lettuces as on Monday.

On Wednesday, Arthur decided to triple the number of lettuces that he had left. At the end of the day there were no lettuces left, and it turned out that he had sold the same number that day, as each of the previous two days.

How many lettuces were sold each day? (2 marks)

5. A rhombus has diagonals of length 420 mm and 560 mm. A circle is drawn inside this rhombus, touching all four sides.

How many millimetres is the circle's radius? (2 marks)

6. Jo set off for a hike along a cross-country trail to Blu Knoll and returned along the same route.

She started at 10:00 am and got back at 4:00 pm, having been up and down hills and along some at ground too.

Her speed along the flat was 4 km/h; and she managed 3 km/h up hills, and 6 km/h down hills. What is the total number of kilometres that Jo walked? (2 marks)

7. On January 1st, Honi was given a large collection of scorpions by her grandmother.

During January, the scorpion population increases by 5%.

From February 1st till the end of February, the population increases by 10% of the population at the beginning of February, and during March it increases by 20% of the population at the beginning of March.

To the nearest whole number, by how many percent has the population increased during the three months? (2 marks)

8. The first two digits of a certain three-digit number form a perfect square and so do the last two digits.

If the number is divisible by 11, what is it? (2 marks)

9. A dingo starts chasing a rabbit which is 15 dingo leaps in front of the dingo.

He (the dingo) takes 5 leaps while she (the rabbit) takes 6, but he covers as much ground in two leaps as she does in three.

How many leaps will it take the dingo to catch the rabbit? (3 marks)

10. Five consecutive positive integers have the property that the sum of the squares of the three smallest is equal to the sum of the squares of the two largest.

What is this common sum? (3 marks)

11. Jasper and Mariko have to travel 12 km to get home but have only one bicycle between them.

Their travel plan is such that they start and finish together. Jasper sets out on the bicycle at 10 km/h, then leaves the bicycle and walks on at 4 km/h.

Mariko sets out walking at 5 km/h, reaches the bicycle and rides home at 8 km/h.

For how many minutes was the bicycle not in motion? (3 marks)

12. For full marks explain how you found your solution.

Let ABCD be a rectangle, and let E be a point on BC and F a point on CD such that BE = DF and AEF is an equilateral triangle.

Prove that the area of triangle ECF equals the sum of the areas of triangles ABE and AFD. (3 marks)