IMO 2001, Problem 2

Let

a

b

c

be positive real numbers. Prove that

\dfrac{a}{\sqrt{a^2+8bc}} + \dfrac{b}{\sqrt{b^2+8ca}} + \dfrac{c}{\sqrt{c^2+8ab}} \ge 1

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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IMO 2001, Problem 2

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