IMO 1995, Problem 2

This is a proof problem. The answer is an argument, not a number. The solution is deliberately not posted here: reading a solution you didn't fight for teaches almost nothing. On Lemma you attempt it cold, take one of the 3 progressive hints only when genuinely stuck, then compare your proof against a full walkthrough and mark yourself with the same rubric a competition coordinator would use.

Guides that use this problem

• How to prove an olympiad inequality, step by stepMost failed inequality proofs pick a tool in the first minute. The order of attack that picks it last: equality case, degree, substitution, shape, audit.
• Nesbitt's inequality: three proofs and how to spot it in disguiseNesbitt's inequality takes two lines to prove once you know it. The skill is seeing it inside a harder problem, and choosing which of three proofs to use.
• What separates a Level 5 problem from a Level 7, side by sideTwo IMO inequalities, both rated 5 out of 6, sit two rungs apart on the ladder. What actually separates a Level 5 problem from a Level 7, and how to cross.
• When a smoothing argument works, and when it quietly liesSmoothing nudges two variables together and reads the answer off the equal point. Here is the test for when that is valid, and the two ways it fails silently.

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