Maths Olympiad Questions for Years 5 and 6, with Worked Solutions

Many Australian primary schools enter their strongest maths students in the maths olympiad. The problems look short, but each one is a small puzzle: there is no rule to follow, so your child has to find a way in. This guide explains how the olympiad works, which kinds of problems come up most, and works through five sample questions, each solved two ways.

How the primary maths olympiad works

The olympiad is set by the Australasian Problem Solving Mathematical Olympiads (APSMO) and run by each school for its own students, several times through the year. Each olympiad is the same shape:

• Five problems, labelled A to E, getting harder as they go.
• 30 minutes for all five.
• Written answers. There are no choices to pick from, so a guess rarely helps.
• Two divisions: Division J for Years 5 and 6 (and younger students), Division E for Years 7 and 8.

Because the papers are short and come round several times a year, steady practice works better than cramming before one big day.

The problems that come up most

Olympiad problems cover a wide spread, but some kinds turn up again and again. From most to least common:

1. Area and perimeter of stepped, crossed and cut-up shapes.
2. Counting: how many totals, paths, numbers or arrangements.
3. Word problems with unknowns: legs and heads, prices of pairs, sums and differences.
4. Factors, multiples and primes, often with a twist.
5. Digits and place value: how often a digit is written, reversed numbers, palindromes.
6. Cubes and solids: painted cubes, staircases, nets.
7. Missing digits and letter sums.
8. Patterns, remainders and cycles.

Rates, ages, averages, clocks and calendars make up most of the rest. There is very little geometry with angles, and almost no probability.

Five sample questions, each solved two ways

These are written by us in the olympiad’s style, one at each level from A (easiest) to E (hardest). Try each one before reading the solutions.

A. Counting a digit

Zara writes all the whole numbers from 1 to 50. How many times does she write the digit 3?

Method 1 - Solve a simpler problem. Count one place at a time. In the units place, 3 appears in 3, 13, 23, 33 and 43: 5 times. In the tens place, it appears in 30 to 39: 10 times. That is 5 + 10 = 15.

Method 2 - Make an organised list. Write the numbers with a 3: 3, 13, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 43. That is 14 numbers, but 33 has two 3s, so the digit is written 15 times.

Answer: 15. The trap is counting the numbers that contain a 3 (14) instead of the times the digit is written.

B. The amount you cannot make

Stamps come only in 3c and 5c. What is the largest amount you cannot make exactly?

Method 1 - Build a table. Write the amounts in rows of 3. Once an amount in a column can be made, every amount below it can too: just add another 3c stamp. The amounts that never get made are 1, 2, 4 and 7.

Method 2 - Use number properties. 8 = 3 + 5, 9 = 3 + 3 + 3 and 10 = 5 + 5 can all be made. That is three amounts in a row, so adding 3c stamps reaches every amount after them. Nothing above 7 is missing.

Answer: 7c.

C. Round a staircase

A shape has 4 steps, each 2 cm across and 3 cm up, with a flat bottom and a straight right-hand side. What is the distance all the way round it?

Method 1 - Draw a diagram. Push each step across up to the top line, and each step up out to the left-hand side. The shape’s edge now makes a rectangle 8 cm wide and 12 cm high, and pushing the edges did not change their length. So the distance round is 2 × (8 + 12) = 40 cm.

Method 2 - Add every side. Four steps across of 2 cm make 8 cm, and four steps up of 3 cm make 12 cm. Add the bottom (8 cm) and the right-hand side (12 cm): 8 + 12 + 8 + 12 = 40 cm.

Answer: 40 cm.

D. Three prices

An apple and a pear cost 70c together. A pear and a banana cost 50c. An apple and a banana cost 60c. How much do one of each cost altogether?

Method 1 - Use number properties. Add the three pairs: 70 + 50 + 60 = 180c. Each fruit is in two of the pairs, so that is every fruit twice. One of each costs 180 ÷ 2 = 90c.

Method 2 - Guess, check and refine. Guess the apple is 30c. Then the pear is 40c and the banana 30c, but a pear and a banana would be 70c, not 50c. Try 40c: the pear is 30c, the banana 20c, and pear and banana make 50c. It works: 40 + 30 + 20 = 90c.

Answer: 90c.

E. Three darts

A target has rings scoring 1, 4 and 7. Three darts are thrown and every dart hits. How many different totals are possible?

Method 1 - Make an organised list. List the darts from the smallest ring up so no mix is counted twice: 1+1+1, 1+1+4, 1+1+7, 1+4+4, 1+4+7, 1+7+7, 4+4+4, 4+4+7, 4+7+7, 7+7+7. That is 10 mixes, with totals 3, 6, 9, 9, 12, 15, 12, 15, 18 and 21. Some totals repeat, so there are only 7 different totals.

Method 2 - Use number properties. Each ring is 1 plus a multiple of 3 (1, 4 = 1 + 3, 7 = 1 + 6). So every total is 3 plus a multiple of 3, from 3 up to 21: 3, 6, 9, 12, 15, 18, 21. All seven can be made.

Answer: 7. The trap is giving 10, the number of mixes.

The strategies worth learning by name

Olympiad solutions are usually explained through a short list of problem-solving strategies. Children who know them by name have somewhere to start when a problem looks new:

• Make an organised list: write every case in order so none is missed or counted twice.
• Draw a diagram: a sketch, a bar model or a grid often shows the answer.
• Build a table: line up the values and watch them change.
• Work backwards: start from the end and undo each step.
• Find a pattern: try the first few cases and see what repeats.
• Solve a simpler problem: shrink the numbers, solve that, then scale up.
• Guess, check and refine: make a sensible guess and improve it.
• Use number properties: odd and even, multiples, remainders and place value.
• Eliminate possibilities: cross out what cannot be true until one option is left.

How to practise

Past olympiad papers go to schools for their own students and are not free online, so most families practise with problems in the same style. A good routine is one five-problem paper a week, done on paper in 30 minutes, then going back over each problem to see a second way to solve it. Seeing two methods side by side is what builds the habit of looking for a way in.

Our maths olympiad practice gives a fresh five-problem paper every time, for Years 4 to 6, plus a longer holiday paper. Every problem is worked two ways, each under the name of its strategy. It has a pack of its own, and it also comes with the Australian Maths Competition pack, since the same children often sit both. Scholar Tests is not connected with APSMO, and none of our questions comes from an olympiad paper.