AIMO: Australian Intermediate Mathematics Olympiad

The Australian Intermediate Mathematics Olympiad (AIMO) is a four-hour proof-based examination for secondary school students in Australia, run annually by the Australian Maths Trust. Students sit the paper in September, tackling ten problems that require full written solutions rather than multiple-choice answers. The competition targets students in Years 7–10, though eligibility varies by state, and serves as a pathway to the Australian Mathematical Olympiad and international selection.

What AIMO Is

AIMO sits between the Australian Mathematics Competition (AMC) and the Australian Mathematical Olympiad (AMO) in difficulty. Where AMC tests problem-solving across a broad curriculum using multiple-choice and short-answer questions, AIMO demands rigorous proof-writing. You must justify every claim, construct logical arguments, and communicate mathematics clearly.

The Australian Maths Trust introduced AIMO to give younger students experience with olympiad-style problems before they reach the senior AMO. A student who performs well in AIMO develops proof techniques, learns to handle multi-step reasoning, and builds the stamina needed for longer competitions.

AIMO is not a curriculum test. You will not see questions about factorising quadratics or calculating derivatives. Instead, problems explore number theory, combinatorics, geometry, and algebra through puzzles that require insight rather than memorised procedures. A typical problem might ask you to prove that infinitely many integers satisfy a certain divisibility condition, or to count configurations of objects satisfying constraints.

Who Can Enter AIMO

AIMO eligibility depends on your year level and how that maps to your state's schooling system. The competition targets students in Years 7, 8, 9, and 10 under most state systems. In South Australia, where year numbering differs, eligible students are in Years 8–11. In Western Australia, eligibility covers Years 8–10 in the middle school system.

You must be enrolled in an Australian school that has registered for AIMO. The competition is not open to individual entries from home; your school coordinates registration and provides a supervised examination venue. If your school does not currently offer AIMO, ask your mathematics coordinator to contact the Australian Maths Trust.

There is no requirement to have participated in AMC or any other competition first. However, most students who sit AIMO have scored highly in AMC or shown strong interest in problem-solving beyond the classroom.

AIMO Format and Question Style

AIMO consists of ten problems, each worth seven marks, sat over four hours. You write full solutions in an answer booklet, showing all reasoning. Partial credit is awarded for progress toward a solution, so a correct first step or a proof of a special case earns marks even if you do not finish.

Problems increase in difficulty. Questions 1–3 are accessible to most participants with careful thought. Questions 4–7 require stronger technique and often a key insight. Questions 8–10 challenge even the most prepared students; a complete solution to Question 10 is rare.

Consider this example in the style of AIMO Problem 3:

Prove that for any positive integer \( n \), the number \( 2^{2n} - 1 \) is divisible by 3.

A complete solution requires proof, not verification. You might start by checking small cases: \( n = 1 \) gives \( 2^2 - 1 = 3 \), divisible by 3. For \( n = 2 \), \( 2^4 - 1 = 15 \), also divisible by 3. But checking cases is not proof.

The key insight is to factor: \( 2^{2n} - 1 = (2^n)^2 - 1 = (2^n - 1)(2^n + 1) \). Now observe that \( 2 \equiv -1 \pmod{3} \), so \( 2^n \equiv (-1)^n \pmod{3} \). When \( n \) is even, \( 2^n \equiv 1 \pmod{3} \), making \( 2^n - 1 \equiv 0 \pmod{3} \). When \( n \) is odd, \( 2^n \equiv -1 \pmod{3} \), so \( 2^n + 1 \equiv 0 \pmod{3} \). In both cases, one factor is divisible by 3, hence the product is divisible by 3.

This solution demonstrates what AIMO markers expect: clear logic, correct algebra, and explicit justification of each step.

Later problems involve more sophisticated ideas. A geometry problem might require you to construct auxiliary lines and prove similarity. A combinatorics problem might ask you to count arrangements by recursion or generating functions. A number theory problem might involve modular arithmetic, divisibility arguments, or properties of primes.

Dates, Registration, and Fees

AIMO takes place in September each year, typically in the second or third week. The exact date varies annually and is announced by the Australian Maths Trust in the first half of the year. Schools receive notification of the date when registration opens, usually around March or April.

Registration is handled by schools, not individual students. Your school's mathematics coordinator or competition organiser submits entries to the Australian Maths Trust and pays the entry fee on behalf of all participants. The school then typically collects fees from students or covers the cost from the mathematics department budget.

The entry fee changes from year to year. Schools should confirm the current fee with the Australian Maths Trust when registering. The fee covers the examination paper, marking by trained assessors, a certificate, and a results report.

Registration usually closes several weeks before the examination date to allow time for printing and distribution of materials. Late entries may be accepted at the discretion of the Australian Maths Trust, sometimes with an additional fee.

How AIMO Is Scored

Each of the ten problems is worth seven marks, for a maximum total of 70. Markers are experienced mathematicians or mathematics teachers trained by the Australian Maths Trust. They follow a detailed marking scheme that allocates partial credit for progress.

A typical marking scheme might award:

• 1–2 marks for correctly interpreting the problem and making a valid first step
• 3–4 marks for substantial progress, such as proving a special case or establishing a key lemma
• 5–6 marks for a nearly complete solution with minor gaps or errors
• 7 marks for a complete, correct, and clearly communicated solution

Markers look for mathematical rigour. A claim without justification earns no credit, even if the claim is true. If you assert that two triangles are congruent without proving it, you lose marks. If you use a result from outside the expected syllabus without proof, you may lose marks unless the result is elementary.

Conversely, markers reward clear reasoning even when the final answer is wrong. Suppose you make an arithmetic error early in a solution but carry the incorrect value through correctly to reach a conclusion. You will lose marks for the error but earn credit for the subsequent correct reasoning.

The median score on AIMO is typically around 10–15 marks out of 70. A score above 30 is strong. A score above 40 places you among the top performers nationally.

Prizes, Certificates, and Results

Every AIMO participant receives a certificate indicating their score and performance level. The Australian Maths Trust awards medals and prizes to the highest-scoring students, though the exact thresholds vary each year depending on the difficulty of the paper and the distribution of scores.

Typical awards include:

• Medals for the top-scoring students nationally
• High Distinction certificates for scores above a high threshold
• Distinction certificates for scores above a moderate threshold
• Credit certificates for scores above a lower threshold
• Participation certificates for all other entrants

The Australian Maths Trust publishes results several weeks after the examination. Schools receive detailed reports showing each student's score on each problem, along with national statistics. Students can see where they succeeded, where they lost marks, and how their performance compares to the cohort.

High performance in AIMO can lead to invitations to further programs. The Australian Maths Trust uses AIMO results, along with AMC results and other indicators, to identify students for the Australian Mathematical Olympiad and subsequent training programs. Students who excel in AIMO and continue to develop their skills may eventually be selected for the Australian team at the International Mathematical Olympiad.

How to Prepare for AIMO

AIMO preparation differs fundamentally from preparing for a school test. You cannot memorise a list of formulas and expect success. Instead, you must learn proof techniques, develop problem-solving strategies, and build mathematical maturity.

Start by working through past AIMO papers. The Australian Maths Trust publishes past problems, and many are available online with solutions. Attempt each problem seriously before looking at the solution. Spend an hour on a single problem if needed. The struggle to find an approach is where learning happens.

When you read a solution, do not merely verify that it is correct. Ask yourself: What was the key insight? What false starts might I have tried? Could I have discovered this approach myself? Then close the solution and write your own version from memory, filling in all details.

Study the four main olympiad topics:

Number theory: Learn divisibility rules, modular arithmetic, the Euclidean algorithm, properties of primes, and Diophantine equations. Practice proving statements about integers using cases, induction, or contradiction.

Combinatorics: Study counting techniques, the pigeonhole principle, recursion, and combinatorial arguments. Learn to count in multiple ways and to recognise when two counting methods must give the same answer.

Geometry: Master angle chasing, similar triangles, properties of circles, and coordinate geometry. Practice constructing auxiliary lines and recognising common configurations like cyclic quadrilaterals or angle bisectors.

Algebra: Work on inequalities, functional equations, polynomials, and sequences. Learn techniques like factoring, substitution, and bounding.

Books such as Problem-Solving Strategies by Arthur Engel and The Art and Craft of Problem Solving by Paul Zeitz cover these topics at the right level. The Australian Maths Trust also publishes resources specifically for olympiad preparation.

Join or form a problem-solving group at your school. Discussing problems with peers exposes you to different approaches and forces you to articulate your reasoning. Explaining a solution to someone else reveals gaps in your understanding.

Write out full solutions to every problem you attempt, even in practice. Do not settle for "I see how to do it." Writing forces precision. You discover errors in your reasoning, learn to structure arguments, and develop the stamina to write for four hours in the actual examination.

Finally, manage your time in the examination. Read all ten problems in the first fifteen minutes. Identify which problems you can solve and which are beyond your current reach. Spend your time earning marks on problems 1–7 rather than staring at problem 10. A complete solution to problem 5 is worth more than scattered ideas on problem 9.

AIMO vs Other Maths Olympiads

AIMO occupies a specific niche in the Australian mathematics competition landscape. Understanding where it sits helps you choose which competitions to enter and how to allocate preparation time.

The Australian Mathematics Competition (AMC) is much broader, with over 300,000 participants annually. AMC uses multiple-choice and short-answer questions covering school curriculum topics plus problem-solving. It is accessible to students of all levels, while AIMO targets those with strong problem-solving skills.

The Australian Mathematical Olympiad (AMO) is harder than AIMO and restricted to senior students (Years 11–12 in most states). AMO problems are fewer—typically four problems over two days—but each requires deeper insight and longer solutions. Students who perform well in AIMO often progress to AMO in later years.

The Mathematics Challenge for Young Australians (MCYA) runs throughout the year and includes multiple stages: the Challenge, Enrichment, and Olympiad stages. The Olympiad stage of MCYA is comparable in difficulty to AIMO and serves a similar purpose, though the format and timing differ.

Internationally, AIMO resembles the first round of olympiad programs in other countries. It is similar in style to the United Kingdom Mathematics Trust's Intermediate Mathematical Olympiad or the earlier rounds of the USA Mathematical Olympiad pipeline. Students who excel in AIMO and continue developing their skills may eventually compete in the International Mathematical Olympiad, where Australia has a strong tradition.

Official Resources and Past Results

The Australian Maths Trust maintains the official AIMO website and publishes past papers, solutions, and results. Schools and students can access these resources through the Trust's online portal.

Past AIMO papers are invaluable for preparation. Solutions are written by the problem authors and markers, showing the intended approach and the level of detail expected. Some problems have multiple solutions, and the published solutions often discuss alternative approaches.

The Australian Maths Trust also publishes statistics each year showing the distribution of scores, the award cut-offs, and the number of participants. These statistics help you calibrate your preparation and set realistic goals.

For students seeking additional support, the Australian Maths Trust runs workshops and online courses covering olympiad problem-solving. These programs are taught by experienced olympiad trainers and past International Mathematical Olympiad participants.

Teachers preparing students for AIMO can access professional development resources through the Trust, including marking guides, teaching notes, and problem-solving workshops.

Results are typically released in October or November, several weeks after the September examination. Schools receive detailed reports, and students can access their individual results through the Australian Maths Trust portal. The reports include your score on each problem, national percentile, and award level.

High-performing students may receive invitations to further programs based on their AIMO results. The Australian Maths Trust uses AIMO performance as one input into selection for the Australian Mathematical Olympiad and subsequent training camps. Students who continue to excel through these programs may eventually be selected for the Australian team at the International Mathematical Olympiad.

AIMO provides a clear pathway for young mathematicians to develop proof-writing skills, tackle challenging problems, and progress toward national and international competition. Success requires sustained effort, a willingness to struggle with difficult problems, and the discipline to write clear, rigorous solutions. For students who invest this effort, AIMO offers both immediate challenge and long-term opportunity in competitive mathematics.