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Extension & competition maths

Maths challenges and olympiads: past papers, practice and extension problems

Everything a keen CBSE student needs to get started with competition maths: what the UKMT challenges, the AMC 8, AMC 10 and AMC 12 and the maths olympiads are, how to enter, links straight to the official past papers, and 120 original competition-style problems of our own with hints and full solutions.

In India the Mathematical Olympiad route starts with the IOQM. Class 9–10 students are in our Intermediate band and Class 11–12 students in the Senior band; the problems here use NCERT content in the unfamiliar, multi-step way olympiad and IOQM questions do.

Links only We link to the organisers’ official past papers; we don’t host them, and our practice problems are our own, not copies.

Problem of the week

A new problem every Monday, rotating through all three levels.

Practise by level

Forty problems at each level, in six topics. Choose the level for your age, then filter by topic.

Junior ages 11 to 13

Same age band as the UKMT Junior Mathematical Challenge and the MAA AMC 8.

Short problems that need careful thinking rather than advanced content: counting, digits, angles, simple equations and logic puzzles.

40 problems →

Intermediate ages 13 to 16

Same age band as the UKMT Intermediate Mathematical Challenge and the MAA AMC 10.

GCSE / IGCSE / Class 9–10 content used in unfamiliar ways: modular arithmetic, clever counting, circle and area geometry, conditional probability.

40 problems →

Senior ages 16 to 18

Same age band as the UKMT Senior Mathematical Challenge and the MAA AMC 12.

A Level / IB / Class 11–12 ideas: recurrences, inclusion–exclusion, Vieta, expectation, 3D geometry and harder logic.

40 problems →

Practise by topic

Official competitions and past papers

Each link goes to the organiser’s own website. We don’t host their papers, and details such as dates and prices change every year, so check there.

Mathematical Olympiad programme in India (HBCSE, IOQM)

India

The Homi Bhabha Centre for Science Education runs India’s Mathematical Olympiad programme. The first stage is the Indian Olympiad Qualifier in Mathematics (IOQM), with integer-answer questions; it leads to the Regional and Indian National Mathematical Olympiads. HBCSE publishes past papers and solutions.

How to enter: Registration opens each year on the official Olympiad site, usually through a registered centre or school.

UK Mathematics Trust (UKMT) challenges

UK

Three individual Challenges sat in school: Junior (Year 8 and below, England and Wales), Intermediate (Year 11 and below) and Senior (Year 13 and below). Each first round is 25 multiple-choice questions; the Junior and Intermediate papers last 60 minutes and the Senior 90 minutes. High scorers are invited to follow-on rounds: Kangaroo papers and Olympiads with written solutions. Schools enter their students; there is no individual entry.

How to enter: Ask your maths teacher or exams officer: schools register and buy entries through UKMT.

MAA American Mathematics Competitions (AMC 8, AMC 10, AMC 12)

USA and international schools

Run by the Mathematical Association of America. The AMC 8 (grade 8 and below) has 25 multiple-choice questions in 40 minutes. The AMC 10 (grade 10 and below) and AMC 12 (grade 12 and below) each have 25 multiple-choice questions in 75 minutes. Top AMC 10/12 scorers are invited to the AIME and then the USA olympiads. Students take the AMC at a registered school or competition site.

How to enter: A teacher or competition manager registers the school with the MAA; students then sit the paper there.

Australian Mathematics Competition (Australian Maths Trust)

Australia, New Zealand and other schools

The Australian Maths Trust runs the Australian Mathematics Competition for school years 3 to 12: 25 multiple-choice questions followed by 5 integer-answer questions, in 75 minutes for secondary papers. Past papers are sold through the AMT shop.

How to enter: Schools register with the Australian Maths Trust; students enter through their school.

For teachers

Use the problems for a stretch starter, an enrichment club or extension homework. On any problem page, open ‘For teachers’, switch on ‘Add to worksheet’ and pick problems: they go into the same worksheet basket as our question bank, so you can print them from the worksheet builder or set them as homework for a class, with the full solutions as the mark scheme. They are free for every class, including free teacher classes.

Questions

Where can I find UKMT past papers?

On the UK Mathematics Trust’s own website: its competition papers page has free past papers for the Junior, Intermediate and Senior Mathematical Challenges, with solutions, and older papers are sold in the UKMT shop. We link to those pages above. We don’t host UKMT papers or copy their questions.

Where can I get AMC 8, AMC 10 and AMC 12 practice problems?

The Mathematical Association of America publishes sample competitions and sells past AMC papers; its AMC pages are linked above. For extra practice in the same spirit, our 120 original problems are grouped by the same age bands: Junior (AMC 8 age), Intermediate (AMC 10 age) and Senior (AMC 12 age).

Are these real UKMT or AMC questions?

No. Every problem in this section was written by us and is labelled competition-style. None is taken from, or reworded from, a UKMT, MAA AMC, Australian Mathematics Competition or IOQM paper, and each answer is checked independently by a computer program.

How do students enter the maths challenges?

Almost always through their school: UKMT, the MAA and the Australian Maths Trust sell entries to schools, and students sit the paper there. Ask your maths teacher or exams officer. India’s IOQM has its own registration each year on the HBCSE Olympiad site.

Which level should I start with?

Start at the level for your age: Junior for ages 11 to 13, Intermediate for 13 to 16 and Senior for 16 to 18. CBSE students usually start at Intermediate. If the first few feel easy, move up a level.

Can teachers set these problems as homework?

Yes. On any problem page open ‘For teachers’, switch on the ‘Add to worksheet’ buttons and pick problems. They go into your worksheet basket, so you can print them from the worksheet builder or set them as homework, with the solutions as the mark scheme. They are free for every class.