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IMO 1964, Problem 2 · Algebra

IMO 1964, Problem 2

Let aa, bb, cc be the sides of a triangle. Prove that a2(b+c−a)+b2(c+a−b)+c2(a+b−c)≤3abc.a^2(b+c-a) + b^2(c+a-b) + c^2(a+b-c) \le 3abc.
Topic: Algebra Difficulty: 5/6 Expected time: ~45 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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