IMO 1998 SL 9

Origin: MON

Problem

Let a1, a2, . . . , an be positive real numbers such that a1 + a2 + \cdot \cdot \cdot + an < 1. Prove that a1a2 \cdot \cdot \cdot an[1 −(a1 + a2 + \cdot \cdot \cdot + an)] (a1 + a2 + \cdot \cdot \cdot + an)(1 −a1)(1 −a2) \cdot \cdot \cdot (1 −an) \leq nn+1 .

Solution

Set an+1 = 1 −(a1 + \cdot \cdot \cdot + an). Then an+1 > 0, and the desired inequality becomes a1a2 \cdot \cdot \cdot an+1 (1 −a1)(1 −a2) \cdot \cdot \cdot (1 −an+1) \leq nn+1 . To prove it, we observe that 1 −ai = a1 + \cdot \cdot \cdot + ai−1 + ai+1 + \cdot \cdot \cdot + an+1 \geqn n\sqrta1 \cdot \cdot \cdot ai−1ai+1 \cdot \cdot \cdot an+1. Multiplying these inequalities for i = 1, 2, . . ., n + 1, we get exactly the inequality we need.