How to Use Past Papers for Maths Competitions

Editorial overview

Past papers are one of the best resources available to students preparing for maths competitions, but they are also one of the easiest to misuse. Completing a paper, checking the score and immediately moving to the next paper can create the feeling of progress without changing how a student thinks. A better approach treats each paper as evidence: it shows which ideas are familiar, which decisions are slow, which mistakes repeat and which problems require a new way of seeing.

The method should also match the competition. A timed multiple-choice paper such as the UKMT Junior Mathematical Challenge rewards precision, selection and efficient calculation. The Junior Mathematical Olympiad is a two-hour follow-on round with six Olympiad-style questions, so the student must develop written reasoning and proof. Irish Mathematical Olympiad preparation is closer to long-form problem solving, where a correct idea must be turned into a complete argument.

Our view is simple: the value of a past paper is determined less by how many papers a student finishes than by what changes after each paper. One paper analysed carefully can improve a student more than five papers used as disposable tests.

What a past paper can actually teach

A paper can reveal at least five different things:

1. Content gaps: a definition, technique or standard result is not yet secure.
2. Recognition gaps: the student knows the mathematics but does not recognise when it is useful.
3. Execution errors: the plan is sound but arithmetic, notation or casework breaks down.
4. Decision problems: too much time is spent on one question or the wrong questions are attempted first.
5. Communication problems: the answer may be correct, but the reasoning is incomplete or impossible to follow.

These are different problems and need different remedies. Doing another full paper is rarely the best response to all five.

Use different papers for different competitions

UKMT Challenges and Kangaroos

For JMC, the past paper should be used to practise mathematical fluency and question selection. UKMT describes the JMC as a 60-minute, 25-question multiple-choice challenge designed to encourage mathematical reasoning and precision of thought. Students should learn that they are not expected to finish every question: the final questions are intended to be harder, and careful solving is more valuable than random guessing.

Use the paper in three modes:

• Untimed mode: understand the idea and compare multiple methods.
• Selective timed mode: give yourself a short limit for a group of questions and practise moving on.
• Full simulation: reproduce the real time limit, paper conditions and answer-sheet discipline.

For the Junior Kangaroo, the format is still multiple choice, but the questions are intended to stretch students who have performed strongly in the earlier Challenge. Past papers are useful for learning how a problem can be reframed, not for memorising the style of a particular year.

Junior Mathematical Olympiad and other proof-style papers

The Junior Mathematical Olympiad is a follow-on round from the JMC and consists of six Olympiad-style questions over two hours. Its past papers should not be treated like a longer JMC. The key training task is to write a complete solution.

For each problem, first try to identify the central idea without looking at a solution. Then write the argument as if another mathematician must be able to verify every step. A diagram, a pattern or a correct numerical answer is not the same as a proof. Students should practise explaining why a construction works, why a case split is exhaustive, or why a conclusion follows from an invariant.

Irish Mathematical Olympiad and enrichment problems

The Irish Mathematical Olympiad problem archive includes past papers, topic-based problem sets and solution resources. It is especially useful because the archive sits alongside material on geometry, number theory, algebra, inequalities and discrete mathematics.

For IrMO-style work, use past papers to build endurance and flexibility. Do not organise all preparation around a predicted topic list. A student may need to move from an apparently geometric diagram to an algebraic invariant, or from a number-theory observation to a proof by contradiction. Topic practice is helpful for learning tools; mixed papers are necessary for learning when to choose them.

Team competitions

Team papers should be used as communication rehearsals as well as mathematics practice. Assign roles for a round, explain possible approaches aloud, compare incomplete ideas and practise deciding when to abandon a path. A student who can solve a question alone may still struggle to contribute under team time pressure if their reasoning is not communicated clearly.

The four-pass method

Pass 1: Cold attempt

Try the paper without help. Record the start and finish time, but do not turn the exercise into a high-stakes score test too early. For a proof paper, it is often sensible to work untimed first and record how long each problem takes.

Mark questions as certain, uncertain or not started. This is more informative than a single score because it shows whether a correct answer was genuinely understood or reached by a fragile method.

Pass 2: Classify the result

After checking the official solution, classify each missed or uncertain question:

The point of classification is to stop every mistake being labelled “I need more practice”. More practice of the wrong kind can simply strengthen the same bad habit.

Pass 3: Repair the idea

Read the solution actively. Ask:

• What was the first useful observation?
• What information in the question made that observation relevant?
• Which tempting approach was inefficient or impossible?
• Can the solution be expressed using a diagram, a table, an invariant or a simpler case?
• What nearby problem would test whether the idea has transferred?

For multiple-choice problems, do not limit yourself to the official route. Try working backwards from the options, using bounds, eliminating impossible cases or testing a structural pattern. These are legitimate competition skills when used thoughtfully. For proof questions, however, a shortcut that only verifies examples is not a proof.

Pass 4: Re-solve later

Close the solution and return to the problem after a few days. If the student can reproduce the exact steps but cannot explain why they work, the idea has not yet become transferable. Change the numbers, alter one condition or solve the problem by a second method when possible.

This delayed re-solve is where much of the learning happens. Immediate copying creates familiarity; spaced reconstruction tests understanding.

When should students use a timer?

Timing is important, but a timer should be introduced in stages.

1. Learn the method first. Work without a clock until the student can see the structure of the problem.
2. Time a small set. Use 5–8 questions to practise pace without the emotional load of a full paper.
3. Add decision rules. For example: if no useful route appears after two minutes, mark the question and move on.
4. Run full simulations. Use official time limits only after the student has enough technique to benefit from pressure.
5. Review the choices. A slow paper is not always a bad paper; it may show that the student spent time on high-value questions rather than guessing.

In a JMC-style paper, a student should practise scanning and returning to harder questions. In an Olympiad paper, the central timing decision is often whether to continue developing a promising idea or switch to another problem. These are different forms of time management.

How to read a solution without becoming dependent on it

There are three unhelpful extremes: looking at the answer after thirty seconds, refusing to look at any solution, or copying a beautiful solution without reconstructing it. A better sequence is:

1. Make a serious first attempt and write down what you tried.
2. Take a small hint, such as a useful construction or a relevant identity, if available.
3. Return to the problem and continue.
4. Read the full solution only when the next step is genuinely blocked.
5. Close the solution and reproduce the argument in your own words.
6. Record the transferable idea in an error log.

The log should contain triggers, not just labels. “Geometry wrong” is too vague. “When two lengths are related by a midpoint, look for equal triangles or an affine transformation” is more useful, provided the student understands the statement and its limits.

A six-week preparation cycle

For younger students, one focused session of 30–45 minutes may be enough. Older students preparing for proof-based competitions may need longer sessions, but the principle remains the same: analysis and re-solving should take at least as much attention as first attempts.

Common mistakes to avoid

Counting papers instead of learning

“We finished ten papers” says very little. Track recurring errors, questions re-solved successfully and methods that have transferred to new problems.

Starting with the newest paper every time

Save some recent papers for realistic simulations. Older papers are often excellent for learning, and the official UKMT paper archive provides questions, solutions and investigations that can be used in a structured way.

Treating all wrong answers as knowledge gaps

A wrong answer may come from a rushed sign, a misunderstood question, an inefficient route or a missing theorem. Diagnosis should come before prescription.

Reading solutions passively

A solution is not a lesson until the student can explain the key decision and use the idea somewhere else.

Forcing proof methods into multiple-choice papers

Multiple-choice competitions allow strategic elimination, estimation and working backwards. Those habits are useful there, but they do not replace proof-writing in an Olympiad setting.

Key Takeaways

• Use past papers as diagnostic tools, not as a paper-counting exercise.
• Match the practice method to the competition format: multiple choice, proof, or team problem solving.
• Separate concept, recognition, execution, timing and proof errors.
• Introduce timed work gradually; speed should be built on understanding.
• Read solutions actively, then close them and re-solve the problem later.
• Keep recent papers for realistic simulations and use older papers for learning and repair.
• The best evidence of progress is that a student can transfer an idea to an unfamiliar problem.
• Official resources such as UKMT papers and solutions and the IrMO problem archive should be part of a wider practice system, not the whole system.