IMO 2000 SL A1

Let a, b, c be positive real numbers with product 1. Prove

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IMO 2000 SL A1

Origin: USA | Category: Algebra

Problem

Let a, b, c be positive real numbers with product 1. Prove that  a −1 + 1 b   b −1 + 1 c   c −1 + 1 a  \leq1.

Solution

Elementary computation gives  a −1 + 1 b   b −1 + 1 c  = ab−a+ a c −b+ 1 −1 c + 1 −1 b + 1 bc. Using ab = 1 c and bc = a we obtain  a −1 + 1 b   b −1 + 1 c  = a c −b −1 b + 2 \leqa c , since b + 1 b \geq2. Similarly we obtain  b −1 + 1 c   c −1 + 1 a  \leqb a and  c −1 + 1 a   a −1 + 1 b  \leqc b. The desired inequality follows from the previous three inequalities. Equal- ity holds if and only if a = b = c = 1.