Proof Writing and Contest Strategy: IOQM, RMO and INMO

Weightage: This chapter covers how the mathematics olympiad is structured and how to approach it. The pathway and formats below are as reported in recent cycles, so confirm dates, cut-offs and rules on the official HBCSE and Mathematics Olympiad Foundation notices, which change from year to year.

1. The pathway

The mathematics route in India runs in stages: the Indian Olympiad Qualifier in Mathematics (IOQM), then the Regional Mathematical Olympiad (RMO), then the Indian National Mathematical Olympiad (INMO). Top INMO performers are invited to training camps, and the final team of six represents India at the International Mathematical Olympiad.

Each stage is a filter. Qualification to the next stage depends on a rank-based or score-based cut-off set each year, with separate lists by region and class category, so treat any published number as a past value rather than a promise.

2. The IOQM format

As reported for recent years, the IOQM has 30 questions with integer answers from 00 to 99. Ten questions carry 2 marks, ten carry 3 marks and ten carry 5 marks, for 100 marks in 3 hours, with no negative marking.

Three consequences follow:

• No negative marking, so never leave an answer blank. A guess costs nothing.
• The answer must be exact. A fraction, a surd or a three-digit number is not accepted, so the problem is designed to end in two digits, often by taking a remainder.
• Marks are tilted to the hard end. The last ten problems carry half the marks, so a fast start on easy problems matters less than finishing a few of the 5-mark ones.

3. The RMO and INMO format

These are proof-based papers. As reported, the RMO has a small set of problems in three hours, and the INMO has six problems over several hours. The number of problems, their marks and the duration are fixed by the notice for the year.

Problems usually span the four areas of number theory, algebra, combinatorics and geometry, and the INMO pairs them in a way that a student strong in only two areas will struggle. Each solution is graded by a human marker against a scheme with partial credit.

4. What a good proof looks like

A marker reads many scripts quickly, so structure is a courtesy and a defence.

1. Restate what you will prove, in one line, with the notation defined.
2. Give the idea in a sentence before the detail, such as "we colour the board and count".
3. Justify every step that is not obvious, and name the theorem you use with its hypothesis checked.
4. Cover all cases, including small values, zero, negatives and degenerate figures.
5. Conclude explicitly, stating that the original claim follows.

A model write-up. Claim: the sum of three consecutive cubes is divisible by 9.

Proof. Let the numbers be $n-1$, $n$ and $n+1$. Then $(n-1{)}^3+n^3+(n+1{)}^3=3n^3+6n=3n(n^2+2)$. It is enough to show that $n(n^2+2)$ is divisible by 3. If $3\mid n$ this is clear. Otherwise $n\equiv \pm 1(\mathrm{mod}3)$, so $n^2\equiv 1$ and $n^2+2\equiv 0(\mathrm{mod}3)$. In every case $3\mid n(n^2+2)$, so $9$ divides the sum. $■$

Note what it does: it restates, reduces, splits the cases exhaustively and ends clearly.

5. What costs marks

• Claims without proof, such as "clearly" or "it is easy to see" on a step that is the heart of the problem.
• Checking small cases and calling it a proof.
• A circular argument, assuming what you are asked to show.
• An unjustified diagram, treating a configuration that merely looks right as true.
• Messy cancelling, such as dividing by a quantity that may be zero.

A partial solution still earns credit. Proving a special case, finding the right invariant or establishing a key lemma are all marked, so write down what you can prove even if you cannot finish.

6. Time and attention

For the IOQM, spend the first 30 to 40 minutes on the 2 and 3 mark problems to bank marks, then give most of the remaining time to the 5-mark problems, answering them in the order of confidence. Leave the final ten minutes to enter answers cleanly and fill every blank.

For the RMO and INMO, read all problems first, pick the one you can start, and work on it until stuck, then switch. A good split is to aim to solve a few problems fully rather than touching all of them, since complete solutions score far more than fragments. Reserve time at the end to rewrite and check one solution well.

7. A practice routine

Problem-solving is a skill built by struggle, not by reading.

• Spend 30 to 45 minutes on a problem before looking at a hint. The struggle is the learning.
• Keep a log. After each problem, write the technique that cracked it and the idea you missed.
• Learn the standard toolkit in each area, as in the earlier chapters, then use past papers to test it.
• Write solutions in full at least weekly, and have someone more experienced read them.
• Mix timed and untimed practice. Untimed work builds depth, timed work builds speed.

Past IOQM, RMO and INMO papers with official solutions are the best material. Work them by area first, then as full mock papers.

8. Beyond India

The camps that follow the INMO prepare the team for the International Mathematical Olympiad, where six problems are set over two days. Selection and camp structure are decided by the organisers, so follow the official announcements.

Common traps

• Leaving IOQM answers blank in a paper with no negative marking.
• Giving an answer outside the 00 to 99 range by forgetting the final reduction.
• Spreading effort over every proof problem and finishing none.
• Writing a proof only for the example that worked.
• Reading solutions too early.

Memory aids

• "Restate, idea, justify, cases, conclude": proof shape.
• "No blanks": the IOQM rule.
• "Finish few, not touch all": proof paper time.

Summary

The mathematics olympiad is a filter of three stages, from the integer-answer IOQM through the RMO to the proof-based INMO, with an international team at the top. The IOQM rewards exact answers and fills every blank, while the RMO and INMO reward complete, justified proofs.

Good preparation is a toolkit in four areas, a habit of writing full solutions, and a log of the techniques that worked.

Exam protocol

• IOQM: bank the early marks, attempt the 5-mark problems, fill every answer.
• RMO and INMO: read everything, choose, finish a few problems fully.
• Write each proof with a restatement, an idea, justified steps and a conclusion.
• Confirm all dates and rules on the official notice.