INMO Previous Year Papers

INMO Previous Year Papers

INMO (Indian National Mathematical Olympiad) is the third stage and the national pinnacle of India's Mathematical Olympiad programme for any student aiming to represent India at the International Mathematical Olympiad (IMO). Conducted by the Mathematics Olympiad Cell at the Homi Bhabha Centre for Science Education (HBCSE) under the National Board for Higher Mathematics (NBHM), INMO has been held every year since 1989. Around 900 students qualify from RMO to write INMO nationally, and of these, only around 35 to 40 students become INMO Awardees and are invited to the International Mathematical Olympiad Training Camp (IMOTC). Solving INMO previous papers helps you understand the depth of proof expected, the level of mathematical creativity required, and the topic weightage that has emerged over decades of the exam.

INMO is a step significantly harder than RMO. The problems are fewer but deeper, the time per question is greater, and the proof-writing standard expected is far closer to that of the IMO. Download the INMO previous papers PDF to practise at your own pace, or use them as timed 4.5-hour sessions under real exam conditions with no references and no calculators.

How to Download INMO Previous Papers PDF

Getting your INMO previous papers PDF from the official sources is free and straightforward. Just follow these simple steps.

1. Visit the official HBCSE Olympiads portal at olympiads.hbcse.tifr.res.in.
2. Navigate to the Past Papers section under the Mathematical Olympiad programme.
3. Click the link for the year you want to practise and save the PDF on your laptop.
4. Print the paper and set a strict 4.5-hour timer before you begin.
5. Attempt all 6 questions in writing with complete proofs, under real exam conditions and with no references.
6. After finishing, compare your solutions against the official HBCSE solutions published on olympiads.hbcse.tifr.res.in, and review community solutions on Art of Problem Solving (artofproblemsolving.com) for alternate approaches.

INMO papers from 2015 to 2025 are available on the HBCSE Olympiads portal. Pre-2015 papers are also accessible in the public domain through AoPS and community archives. Unlike RMO, the INMO paper itself was held in its standard proof-based format throughout the COVID years (2021 and 2022), so there are no gaps in the paper archive. However, the qualification pipeline was significantly altered during this period: the first two stages, PRMO and RMO, were merged into a single exam called IOQM (Indian Olympiad Qualifier in Mathematics), from which students qualified directly to write the standard INMO.

Benefits of Solving INMO Previous Years Papers

Working through past INMO papers gives you advantages no other resource can match.

• Understand IMO-level proof depth: INMO problems demand proofs of a significantly higher standard than RMO. Past papers show you exactly what level of rigour and elegance is expected for a full-mark solution.
• Calibrate time per question: INMO gives you 4.5 hours for 6 problems, roughly 45 minutes per question. Practice teaches you when to commit to a problem, when to take a fresh approach, and when to move on and return.
• Identify topic patterns across decades: Geometry, Algebra, Number Theory, and Combinatorics each appear in consistent proportions. Solving papers from multiple years reveals which problem types and techniques repeat at this level.
• Strengthen proof elegance: At INMO level, a clumsy but correct proof scores less than a clean, well-structured one. Each paper you attempt and review improves both the substance and the presentation of your solutions.
• Build the stamina for IMOTC selection: Writing 6 high-difficulty proofs over 4.5 hours demands extraordinary concentration. Only repeated timed practice under real conditions builds this stamina.
• Understand the gap between RMO and INMO: Comparing your RMO performance with your INMO paper attempts gives you a precise measure of how much more depth you need to develop in each topic.

Pair your paper-solving with detailed solution review from the HBCSE portal and the Art of Problem Solving community for the best results.

INMO Exam Pattern & Marking Scheme

Understanding the INMO exam pattern is the first step before solving any paper. INMO is an offline, pen-and-paper exam requiring fully written mathematical proofs. It is conducted simultaneously at centres across the country on the third Sunday of January every year.

INMO Exam Pattern

The table below shows the INMO exam structure. All six problems carry equal marks and require complete descriptive solutions. There are no multiple choice options and no calculators are allowed.

There are no sectional time limits. Candidates attempt questions in any order within the 4.5-hour window. Unlike RMO, which is conducted in multiple regional languages, INMO is conducted only in English. The INMO paper has been held every year since 1989. During the COVID years (2021 and 2022), the qualification pathway was significantly altered (PRMO and RMO were merged into IOQM), but INMO itself continued in its standard proof-based format.

INMO Marking Scheme

The table below gives an approximate sense of how marks are distributed in INMO. Each problem is evaluated holistically by examiners for mathematical rigour and core structural insights, rather than against a rigid step-by-step point grid. The ranges below are indicative of the level of progress expected at each band.

There is no negative marking in INMO. Unlike a rigid step-by-step marking grid, each problem is evaluated holistically by examiners who assess the mathematical rigour and core structural insights in your solution. Partial credit is heavily awarded for solutions that demonstrate significant progress and genuine understanding of the key idea, even when the proof is not fully complete. Writing down every relevant observation, partial construction, or special case you can, even without a complete argument, can meaningfully contribute to your score, and is often the difference between becoming an INMO Awardee and not.

INMO Exam Analysis Over the Years

A year-wise look at INMO helps you plan smarter.

The INMO exam analysis shows that Geometry is consistently among the hardest problems and is often the most decisive for IMOTC selection. Algebra at INMO level moves beyond standard techniques into functional equations, inequalities, and structural analysis. Number Theory problems require deep familiarity with modular arithmetic, divisibility arguments, and bounding methods. Combinatorics problems test creative logical reasoning rather than case-by-case enumeration. Students who can score 12 or more on at least three problems are typically in contention for becoming INMO Awardees.

INMO Preparation Tips

Smart preparation beats long preparation. Use these INMO preparation tips to plan your study schedule.

• Solve past INMO and ISL (International Shortlist) problems by topic: The International Mathematical Olympiad Shortlist is the single best source of INMO-level problems. Organise them by topic and work through them systematically.
• Master Geometry at the olympiad level: Power of a point, radical axes, inversion, projective geometry, and trigonometric cevians are all required at INMO. Problem-solving fluency in these areas is non-negotiable.
• Develop functional equation techniques: Cauchy's equation, substitution strategies, injectivity and surjectivity arguments, and parity checks are core algebra tools at this level.
• Build Number Theory beyond the basics: Lifting the Exponent Lemma (LTE), Zsygmondy's theorem, Hensel's lemma, and quadratic reciprocity appear regularly. Work through problems that require combining multiple NT tools.
• Study Combinatorics proof methods: Double counting, generating functions, bijections, and extremal principles are the key techniques. Avoid case-bashing; INMO combinatorics rewards structural insight.
• Write every solution in full: Never settle for understanding the approach mentally. Write the complete proof on paper for every problem you solve, and then compare with a model solution to find gaps in rigour or presentation.
• Use Art of Problem Solving for community feedback: AoPS threads on INMO problems often contain multiple solution approaches, common mistakes, and scoring discussions that are invaluable for self-assessment.

For beginners qualifying RMO for the first time, focus on closing the gap between RMO-level and INMO-level by working through one INMO paper per month and deeply studying solutions. Repeat aspirants should target specific topics where they have historically not reached double-digit scores and study those through ISL problems.