15 Classic Maths Olympiad Problems With Solutions

The best way to understand what a maths olympiad actually asks is to sit down with real problems. Below are fifteen classic questions, arranged from Class 3 level upwards, each with a full solution and — more importantly — the idea behind it.

Work through these with your child. Let them try first; the struggle is where the learning happens.

Warm-ups (Classes 3–4)

1. The missing number What comes next: 2, 6, 12, 20, 30, ___?

Solution: The differences are 4, 6, 8, 10 — increasing by 2. Next difference is 12, so the answer is 42. Idea: When a series looks irregular, look at the differences.

2. Counting sevens How many times does the digit 7 appear when writing 1 to 100?

Solution: In the units place: 7, 17, 27 … 97 = 10 times. In the tens place: 70 to 79 = 10 times. Total 20. Idea: Count each digit position separately.

3. The rope A rope 48 m long is cut into 6 m pieces. How many cuts?

Solution: 48 ÷ 6 = 8 pieces, but the last piece needs no cut. 7 cuts. Idea: Pieces and cuts differ by one. This "fencepost" idea appears constantly.

4. Doubling A number is doubled and then 6 is added, giving 38. What was it?

Solution: Work backwards. 38 − 6 = 32, then ÷ 2 = 16. Idea: Reverse the operations in reverse order.

5. Sum to twenty What is 1 + 2 + 3 + … + 20?

Solution: Pair them: 1 + 20, 2 + 19, … ten pairs of 21 = 210. Idea: Pairing from the ends is Gauss's trick and works for any consecutive run.

Building up (Classes 5–6)

6. Three consecutive numbers Their sum is 51. What is the largest?

Solution: The middle number is 51 ÷ 3 = 17, so the numbers are 16, 17, 18. Largest = 18. Idea: For an odd count of consecutive numbers, the average is the middle one.

7. Rectangles in a grid How many rectangles (including squares) are in a 2 × 2 grid of squares?

Solution: Choose 2 of the 3 vertical lines and 2 of the 3 horizontal lines: 3 × 3 = 9. Idea: A rectangle is defined by two pairs of lines, not by counting shapes one by one.

8. Days ahead If today is Tuesday, what day is it after 100 days?

Solution: 100 ÷ 7 leaves remainder 2. Two days after Tuesday is Thursday. Idea: Cyclic questions are remainder questions.

9. The certain blue A box has 5 red and 3 blue balls. How many must you take out to be sure of getting a blue one?

Solution: Worst case you draw all 5 red first, then a blue. 6 balls. Idea: "To be sure" always means assume the worst possible order.

10. Clock angle What is the angle between the hands at 3:00?

Solution: Each hour mark is 30°. Three marks apart = 90°. Idea: The full circle is 360° over 12 hours, so 30° per hour.

Genuine olympiad level (Classes 7–8)

11. Last digit What is the units digit of 7¹⁰⁰?

Solution: The units digits of powers of 7 cycle: 7, 9, 3, 1, then repeat every 4. 100 ÷ 4 leaves remainder 0, so we take the fourth in the cycle: 1. Idea: Units digits always cycle. Find the cycle length, then use the remainder.

12. Handshakes Ten people each shake hands with everyone else once. How many handshakes?

Solution: Each of 10 people shakes 9 hands = 90, but every handshake is counted twice. 90 ÷ 2 = 45. Idea: Divide by two whenever you have counted each pair from both ends.

13. Two numbers Two numbers add to 20 and multiply to 96. Find them.

Solution: Try factor pairs of 96: 8 × 12 = 96 and 8 + 12 = 20. So 8 and 12. Idea: For small numbers, listing factor pairs beats forming a quadratic.

14. Diagonals How many diagonals does a hexagon have?

Solution: Each of 6 vertices connects to 3 non-adjacent vertices = 18, halved for double counting = 9. Idea: The same divide-by-two principle as handshakes.

15. The pigeonhole Thirteen people are in a room. Show that at least two share a birth month.

Solution: There are 12 months. If each of the first 12 people had a different month, the thirteenth must repeat one. Proved. Idea: If you have more items than boxes, some box holds two. Simple, and surprisingly powerful.

What these problems have in common

Notice how few required heavy calculation. What they needed was:

That is the real olympiad syllabus. Arithmetic gets you through the paper quickly; these ideas get you the marks.

Frequently asked questions

How should my child use these problems? Attempt each one for five minutes before looking at the solution. Reading solutions without struggling first teaches very little.

What if they get almost all of them wrong? That is normal on first exposure. The ideas are unfamiliar, not hard. Re-attempt the same fifteen a week later and the difference is usually striking.

Are these the actual difficulty of an olympiad paper? Problems 1 to 10 are typical of the main sections; 11 to 15 are closer to the achievers section.

Should we move on to past papers next? Once your child can solve most of these unaided, yes. Before that, more practice with ideas like these is a better use of time.

Does mental maths help with these? Indirectly but importantly — it frees up attention for the reasoning. A child still working out 7 × 9 has less capacity for spotting a pattern.

Ready for structured preparation?

Our Maths Olympiad course for Grades 3 to 8 teaches exactly these ideas, with graded practice and mock tests.

Book a free demo class and let us assess where your child is.

Cover image: Blue Sonic — CC0.