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IMO 1983, Problem 6 · Algebra

IMO 1983, Problem 6

Let aa, bb, cc be the side lengths of a triangle. Prove that a2b(a−b)+b2c(b−c)+c2a(c−a)≥0a^2b(a-b) + b^2c(b-c) + c^2a(c-a) \ge 0, and determine when equality holds.
Topic: Algebra Difficulty: 6/6 Expected time: ~60 min Progressive hints on Lemma: 3

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.

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