IMO 1987 SL 6

Origin: GRE

Problem

Show that if a, b, c are the lengths of the sides of a triangle and if 2S = a + b + c, then an b + c + bn c + a + cn a + b \geq 2 n−2 Sn−1, n \geq1.

Solution

Suppose w.l.o.g. that a \geqb \geqc. Then 1/(b + c) \geq1/(a + c) \geq1/(a + b). Chebyshev’s inequality yields an b + c + bn a + c + cn a + b \geq1 3(an + bn + cn)  b + c + a + c + a + b  . (1) By the Cauchy-Schwarz inequality we have

2(a + b + c)  b + c + a + c + a + b  \geq9, and the mean inequality yields (an + bn + cn)/3 \geq[(a + b + c)/3]n. We obtain from (1) that an b + c + bn a + c + cn a + b \geq a + b + c n  b + c + a + c + a + b  \geq3 a + b + c n−1

2 n−2 Sn−1.