Proof Writing and Contest Strategy: IOQM, RMO and INMO — Practice

Question 1

What are the answer range and negative marking rule in the IOQM, as reported for recent years?

Accepted Answer

Integer answers from 00 to 99, with no negative marking.

Question 2

Name the stages of the mathematics olympiad in India in order.

Accepted Answer

IOQM, then RMO, then INMO, followed by the training camps for the international team.

Question 3

Prove that the sum of three consecutive cubes is divisible by 9.

Accepted Answer

(n-1)^3 + n^3 + (n+1)^3 = 3n^3 + 6n = 3n(n^2 + 2). If 3 divides n this is clear. Otherwise n^2 is 1 mod 3 so n^2 + 2 is 0 mod 3. So 3 divides n(n^2 + 2) and 9 divides the sum.

Question 4

Describe a time plan for the 3-hour IOQM.

Accepted Answer

Spend 30 to 40 minutes on the 2 and 3 mark problems, then most of the remaining time on the 5-mark problems in order of confidence, and keep ten minutes to enter answers and fill every blank.

Question 5

Name three things that cost marks in a proof.

Accepted Answer

Unproved claims at the key step, small cases offered as a proof, and circular reasoning.

Question 6

Prove that n^2 + n + 1 is odd for every integer n.

Accepted Answer

n^2 + n = n(n + 1) is a product of consecutive integers, so it is even. Adding 1 gives an odd number.

Question 7

On a proof paper you have partial progress on three problems and nothing complete. What should you do with the last hour?

Accepted Answer

Pick the problem closest to completion, finish it fully, and write it carefully, since complete solutions score far more than fragments. Write down proven lemmas on the others for partial credit.

Question 8

How long should you struggle with a problem before looking at a hint?

Accepted Answer

About 30 to 45 minutes.

Practice — Proof Writing and Contest Strategy: IOQM, RMO and INMO

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