How to Prepare for Maths Olympiads

A Maths Olympiad is a mathematical competition designed to test deeper:

Instead of telling students exactly which method to use, Olympiad problems often require them to discover the method themselves. A student may understand every mathematical concept in a question and still find the problem difficult. That is because the real challenge is often: knowing what to do with the mathematics.

School mathematics and Olympiad mathematics overlap, but their emphasis can be very different.

Strong school mathematics provides the foundation, but Olympiad preparation develops a different level of thinking.

Students do not have to be Olympiad winners before beginning. What matters is starting at an appropriate level.

A common mistake is starting immediately with extremely advanced Olympiad problems. If a student cannot make meaningful progress on most questions, they may simply become dependent on solutions.

A better progression is: School Foundations → Challenge Problems → Advanced Competition Questions → Olympiad Problems.

The difficulty should increase gradually.

Before creating a preparation plan, identify:

A student who lacks algebra fluency needs a different plan from a student who understands algebra but struggles to choose strategies.

Olympiad problems may be creative, but they still rely on core mathematics. Students should be secure in areas appropriate to their age, including:

Weak foundations make advanced problem solving unnecessarily difficult.

Learning topics several years ahead does not automatically make a student good at Olympiads. Olympiad success depends heavily on:

A relatively simple mathematical idea can produce a very difficult Olympiad problem.

At higher levels, Maths Olympiad problems commonly draw from areas such as:

Students do not need to master everything immediately.

Number theory is one of the most important Olympiad areas. Useful concepts include:

Students should learn to recognise structure in numbers rather than relying only on calculation.

Parity means whether an integer is Even or Odd. It sounds simple, but parity can solve surprisingly difficult problems.

Students should understand relationships such as Even + Even = Even, Odd + Odd = Even, Odd + Even = Odd, and what happens under multiplication.

Olympiad problems often hide useful information inside these basic properties.

Students should move beyond memorising divisibility tests.

This turns number theory into reasoning rather than arithmetic.

Olympiad algebra is not only about solving standard equations. Students may need to:

Students should become comfortable seeing the same expression in several different forms.

In school algebra, students sometimes develop the habit of expanding expressions as soon as they see brackets. In Olympiad problems, factorised structure may contain useful information.

Before expanding, ask: does keeping this expression factorised reveal something important?

Good problem solvers choose the representation that makes the mathematics easier to understand.

Olympiad geometry often tests relationships rather than formula recall. Students may need to use:

A diagram may contain several hidden relationships.

One of the simplest ways to improve geometry performance is to draw clearly. Students should:

A useful extra line can completely change a geometry problem.

Combinatorics involves counting possibilities carefully. Students may encounter:

A major challenge is avoiding double counting. Students should learn to organise possibilities systematically.

This reasoning is more valuable than memorising many counting formulas.

At advanced Olympiad levels, simply giving the answer may not be enough. Students need to demonstrate why the result must be true.

Each important conclusion should follow from something already established.

Students should not wait until advanced Olympiads before learning mathematical communication.

Even after solving a simpler problem, ask: can you explain why your answer must be correct? This gradually develops proof-writing habits.

Showing that something works for 1, 2, 3, 4 and 5 does not necessarily prove it works for every possible number.

Examples can help discover a pattern. Proof explains why the pattern must continue.

This distinction becomes increasingly important in Olympiad mathematics.

Students should gradually learn several strategies. Useful Olympiad strategies include:

The aim is not to force every problem into one method. It is to have several possible approaches available.

If a problem involving 100 objects looks impossible, investigate what happens with 1, 2, 3, 4 or 5 objects.

that are difficult to see in the full problem.

But remember: finding a pattern is not the same as proving it.

Some problems are easier when approached from the required result.

Ask: what would need to be true immediately before the final condition? Continue working backwards until the connection to the starting information becomes clear.

Instead of solving every possibility at once, divide the problem into logical cases.

For example: Case 1: Number is even, Case 2: Number is odd, or Case 1: Point lies inside, Case 2: Point lies outside.

Cases must cover all possibilities without unnecessary overlap.

Symmetry can reduce the amount of work dramatically.

Recognising symmetry is a powerful Olympiad skill.

An invariant is something that remains unchanged while a process occurs.

If one property never changes, it can reveal what final states are possible or impossible.

Sometimes the simplest route is to assume the opposite of what needs to be proved. Then show that this assumption creates an impossibility.

The structure is: Assume Opposite → Follow Logic → Reach Contradiction → Original Statement Must Hold.

This becomes particularly useful in proof-based Olympiad mathematics.

possible case. Extreme elements can sometimes reveal hidden restrictions.

A different representation can make the structure much easier to see.

This is one of the most important Olympiad habits.

Avoid: Difficult Question → 2 Minutes Thinking → Read Solution.

Olympiad ability develops during the period when a student is genuinely searching for an idea. Students should make a meaningful attempt first.

Before opening the solution:

Only then consider a hint or solution.

When possible, use progressively stronger hints.

This preserves more of the problem-solving process.

Solutions should be studied actively. Do not simply think “That makes sense.”

A powerful learning method is: Attempt → Review → Wait → Reattempt.

After understanding a solution, wait several days. Then try the same problem again from the beginning.

If the student still needs the solution, the idea has not yet become secure.

Over time, this becomes a personalised problem-solving resource.

Not every wrong answer has the same cause.

Identifying the error type makes revision more effective.

Past competition problems are extremely valuable. But avoid: Paper → Score → New Paper → Score → New Paper.

Instead use: Attempt → Mark → Analyse → Learn → Reattempt → Transfer.

Students should understand why they missed each significant problem.

If a student is weak in geometry, repeatedly completing full Olympiad papers may not fix the weakness. Instead:

This is more targeted.

Topic practice is useful, but students also need situations where the topic is not labelled.

In a real Olympiad, the paper does not say “Use Number Theory.” Students must decide that themselves.

Mixed practice develops: Strategy Selection.

Do not make every Olympiad session timed from the beginning. First develop Reasoning, then Accuracy, then Efficiency, then Competition Timing.

A useful progression is: Untimed Problems → Timed Sets → Timed Sections → Full Competition Papers.

Olympiad persistence is valuable, but competition strategy also requires judgement.

During a timed paper: Try the problem → Make meaningful progress → If completely stuck, move on → Return later.

Spending most of the competition on one unsolved question may not be the best strategy.

Students sometimes believe Olympiad preparation means becoming faster. Speed matters later.

Initially, the priority should be: Correct Reasoning. A fast incorrect solution has little value.

Build speed only after the underlying approach is reliable.

Strong mental Maths can reduce unnecessary working and free attention for harder reasoning. Students can develop:

The aim is not speed for its own sake. It is reducing cognitive effort on routine calculations.

Students should move beyond “I know this rule” towards “I understand why this rule works.”

Olympiad mathematics rewards deeper understanding.

When possible, solve the same problem differently — for example Algebraically, Geometrically, By Counting, Using Symmetry.

Comparing approaches develops mathematical flexibility.

Teaching a solution can expose gaps that silent reading hides.

There is no universal ideal number of hours. Consistency matters more than excessive volume.

For many students, regular focused sessions are more useful than occasional marathon practice.

An hour spent deeply analysing difficult problems may be more productive than several hours of rushed work.

This is an example only. The programme should be adapted to the student's age, competition and school workload.

Do not attempt to learn every advanced Olympiad topic in a few weeks.

Parents do not need advanced mathematical knowledge to help.

A useful question is “What approach have you tried?” rather than “Why can't you solve it?”

Maths Olympiads should encourage intellectual curiosity.

Students should not believe every question must be solved, every competition must result in a certificate, or every score determines mathematical ability.

Difficult problems are deliberately difficult. Learning to handle them is part of the purpose.

Frustration often occurs when students are used to getting normal school questions correct quickly.

Help them understand: struggling is normal in Olympiad mathematics. Then reduce difficulty temporarily.

A progression of achievable challenges can rebuild momentum.

Challenge should continue to increase.

These indicate genuine mathematical development.

How early should students begin Maths Olympiad preparation?

It depends on their current level and target competition. Long-term development is usually more effective than last-minute preparation because reasoning skills take time to build.

Does a student need to be gifted to prepare for Olympiads?

No. Strong interest, suitable foundations and willingness to work through difficult problems are important.

What topics should students study?

Common areas include number theory, algebra, geometry, combinatorics and proof, although the level depends on the competition.

Should students learn advanced Maths ahead of school?

Not necessarily. Strong reasoning with familiar mathematics can be more useful than rushing through advanced content without depth.

How many Olympiad problems should students solve each day?

There is no fixed number. A small number of carefully analysed problems may be more valuable than many rushed questions.

Should students use solutions?

Yes, but only after a genuine attempt. Solutions should be studied to understand the key idea and then the problem should be reattempted later.

Are past papers enough for Olympiad preparation?

Past papers are valuable but usually work best alongside topic development, strategy practice and error analysis.

How can students improve proof writing?

Practise explaining every important step and ask whether each conclusion has been properly justified.

Can Olympiad Maths help with school Maths?

The deeper reasoning and problem-solving skills developed can support wider mathematical learning, although Olympiad preparation should complement rather than replace curriculum study.

Does Dominis offer Maths Olympiad preparation?

Yes. Support can be tailored according to the student's age, current mathematical level, target competition and problem-solving experience.