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Extension & competition maths

Extension and enrichment maths: challenges, olympiads and problem solving

Everything a keen IB student needs to go beyond the syllabus: 608 original competition-style problems with two hints and a full solution each, ten problem-solving strategy guides, a problem of the week, a route from maths challenges to olympiads and university admissions tests, and links to the official past papers.

Competition problems are a good stretch alongside IB Maths: they reward the reasoning and proof skills that Paper 3 and the internal assessment reward, and they are a common enrichment choice for students aiming at a 7 or at university maths.

Try this week’s problem

Problem of the week

A new problem every Monday, rotating through the levels.

Problem J01

Junior · Number theoryShort answer

How many whole numbers from 1 to 200 have digits that add up to 5?

Hints and full worked solution

Last week’s problem: answer and solution

Each Monday the previous problem’s answer and worked solution appear here.

Practise by level

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.

All free with solutions

165 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.

All free with solutions

173 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.

92 free, 58 with a plan

150 problems →

Olympiad-style ages 15 to 18

In the style of national maths olympiad first rounds.

Fewer, harder problems where the answer is only half the work: you have to find an argument that settles every case. Number theory, inequalities, invariants, extremal arguments and clever counting.

Questions free; solutions with a plan

120 problems →

Practise by topic

TopicJuniorIntermediateSeniorOlympiad-style
Number theory28322224
Combinatorics28302222
Geometry28282218
Algebra28292222
Probability26262010
Logic2728188
Calculus and functions––2416
Extension and enrichment maths: a competition-style problem card with two hints and a worked solution, across Junior, Intermediate, Senior and Olympiad-style levels
Every problem has two hints, a step-by-step solution, why the method works and where the idea leads.

Problem-solving strategies

Stuck on a problem? Each one is tagged with the idea that cracks it. The strategy guides explain each idea with worked examples.

Organised casesCount the oppositeWorking backwardsInvariantsExtremal principlePigeonhole principleParity and remaindersSymmetrySpot the pattern and generaliseProof techniques

Where next

Maths challenges, then olympiad-style problems, then university admissions tests (the TMUA, which Oxford uses from 2026, and STEP). See the route and what each test is.

For teachers

Use the problems for a stretch starter, an enrichment club or extension homework. Every problem has a Project button for the board. The maths club sessions give ready-made 50-minute plans by level and strategy. In December, run Christmaths, our maths advent calendar, as a class competition. On any problem page, open ‘For teachers’, switch on ‘Add to worksheet’ and pick problems to print or set as homework, with the full solutions as the mark scheme.

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.

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

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.

2026–27 dates: Senior Mathematical Challenge: Wednesday 7 October 2026. Senior Kangaroo and British Mathematical Olympiad Round 1: 18 November 2026. BMO Round 2: 20 January 2027. Intermediate Mathematical Challenge: Wednesday 27 January 2027. Grey and Pink Kangaroo and the Cayley, Hamilton and Maclaurin Olympiads: 18 March 2027. Junior Mathematical Challenge: Wednesday 5 May 2027. Junior Kangaroo and Junior Mathematical Olympiad: 15 June 2027. UKMT competition calendar (checked 5 October 2026)

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.

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.

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 608 original problems are grouped by the same age bands: Junior (AMC 8 age), Intermediate (AMC 10 age) and Senior (AMC 12 age), plus an Olympiad-style level.

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.

What is free and what needs a plan?

Free for everyone: this hub, the strategy guides, every Junior and Intermediate problem and most Senior problems with hints and full solutions, the problem of the week and the maths club session plans. Included with every A Level, IB, IGCSE and CBSE plan: hints and solutions for the Olympiad-style and remaining Senior problems, printable club packs and progress saved to your account.

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; Olympiad-style is for students who find Senior comfortable. IB students usually start at Senior. 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. Free problems are free for every class; the others need a plan or school licence.