What is math olympiad? Primary questions with solutions
What math olympiad means, the common primary olympiad question types, and six original questions with step-by-step solutions.
Math olympiad means competition-style maths problems that go beyond the school syllabus and test reasoning and problem-solving strategy. Common primary olympiad question types are patterns, chickens and rabbits, counting, tree planting and fraction tricks, and each has one core method.
Key points
- Olympiad questions test reasoning, not harder syllabus content.
- Each question type has a core method: list, assume, pair or draw.
- Try small numbers first, then look for the pattern.
Where the name comes from
The name comes from the International Mathematical Olympiad (IMO), a world championship for secondary school students. It was first held in Romania in 1959 and is now held every year in a different country.
The IMO runs over two days, with four and a half hours each day to solve three problems. Today, math olympiad also refers to competition-style questions at every level, including primary school.
Common primary olympiad question types
Most primary olympiad questions belong to one of these types. Recognising the type tells you which method to try first.
- Patterns: a sequence or picture follows a rule; find the next term.
- Chickens and rabbits: you know the total number of animals and legs; find how many of each.
- Counting and handshakes: list every case once, without repeats or gaps.
- Gauss sums: pair the first and last numbers to add a long list quickly.
- Tree planting: tell the number of gaps apart from the number of trees.
- Fraction tricks: split fractions so that the middle terms cancel.
Question 1: chickens and rabbits
A cage holds 10 chickens and rabbits with 28 legs in total. How many of each are there?
Suppose all 10 are chickens: that gives 20 legs, 8 fewer than the real total. Each time you swap a chicken for a rabbit you add 2 legs, so you need 8 ÷ 2 = 4 swaps. There are 4 rabbits and 6 chickens. Check: 6 × 2 + 4 × 4 = 28.
Question 2: handshakes
Six students meet and each pair shakes hands once. How many handshakes are there?
The first student shakes hands with the other 5; the second still has 4 people left; and so on. The total is 5 + 4 + 3 + 2 + 1 = 15. Try it with 3 people first (3 handshakes) to check the method.
Question 3: the Gauss sum
Find 1 + 2 + 3 + … + 100.
Pair the first and last numbers: 1 + 100, 2 + 99, 3 + 98 and so on. Each pair adds to 101 and there are 50 pairs, so the total is 101 × 50 = 5050. The arithmetic series formula in secondary school uses exactly this idea.
Read next: Arithmetic and geometric sequences
Question 4: tree planting
Trees are planted every 5 m along a 50 m path, with a tree at both ends. How many trees are there?
50 ÷ 5 = 10, so there are 10 gaps. With trees at both ends there is one more tree than gaps, so the answer is 11. Draw a 10 m path to see why you add 1.
Question 5: splitting fractions
Work out 1/2 + 1/6 + 1/12 + 1/20 + 1/30.
Each denominator is a product of two consecutive whole numbers: 2 = 1 × 2, 6 = 2 × 3 and so on. Since 1/(n × (n + 1)) = 1/n − 1/(n + 1), the sum is (1 − 1/2) + (1/2 − 1/3) + … + (1/5 − 1/6). The middle terms cancel, leaving 1 − 1/6 = 5/6.
Read next: Check it with the fraction calculator
Question 6: find the pattern
What is the next term of 2, 5, 10, 17, 26, …?
The differences are 3, 5, 7 and 9, growing by 2 each time, so the next difference is 11 and the next term is 26 + 11 = 37. Another view: each term is a square number plus 1 (1 + 1, 4 + 1, 9 + 1, …), so the next term is 36 + 1 = 37.
How to start practising
Practise one question type at a time: understand the core method first, then try variations. When stuck, draw a picture, make a table or try smaller numbers until a pattern appears.
Always check your answer, for example by putting it back into the question. Olympiad problems are a challenge on top of school maths, not a replacement for it.
Try one
A cage holds 12 chickens and rabbits with 34 legs. How many of each are there?
Hint: Start by assuming they are all chickens.
Answer: All chickens would give 24 legs, 10 short. Each rabbit adds 2 legs, so there are 10 ÷ 2 = 5 rabbits and 7 chickens. Check: 7 × 2 + 5 × 4 = 34.
Frequently asked questions
What is 奧數 in English?
奧數 is short for Math Olympiad (Mathematical Olympiad). The international competition's full name is the International Mathematical Olympiad, or IMO.
At what age can a child start olympiad maths?
There is no fixed age. A child who is secure with school arithmetic and enjoys puzzles can start with the simpler question types.
Are olympiad questions always hard?
No. Many starter questions need only primary arithmetic; the challenge is finding the method. Once you know the common types, most starter questions can be solved step by step.
Is the IMO for primary pupils?
No. The IMO is for secondary school students, in teams of up to six who must be under 20. Primary pupils take part in other maths competitions at their own level.