IMO 1995 SL A1

Let a, b, and c be positive real numbers such that abc = 1.

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IMO 1995 SL A1

Origin: RUS | Category: Algebra

Problem

Let a, b, and c be positive real numbers such that abc = 1. Prove that a3(b + c) + b3(a + c) + c3(a + b) \geq3 2.

Solution

Let x = 1 a, y = 1 b, z = 1 c. Then xyz = 1 and S = a3(b + c) + b3(c + a) + c3(a + b) = x2 y + z + y2 z + x + z2 x + y . We must prove that S \geq3