NMO Selection Tests for the Balkan and International Mathematical Olympiads

Question

Let n be a positive integer and let x_1, x_2, , x_n be positive real numbers such that x_1x_2 x_n = 1. Prove that _i=1^n x_i^n(1 + x_i) n2^n-1 _i=1^n(1 + x_i).

Accepted Answer

By the power-mean inequality, 1 + a^n (1 + a)^n2^n-1, a 0. (*) Thus, align* _i=1^n x_i^n (1 + x_i) &= _i=1^n x_i^n + _i=1^n x_i^n+1 & _i=1^n x_i^n + n ( _i=1^n x_i )^1+1/n (AM-GM) &= _i=1^n x_i^n + n = _i=1^n (1 + x_i^n) & 12^n-1 _i=1^n (1 + x_i)^n by (*) & n2^n-1 _i=1^n (1 + x_i). (AM-GM) align* Clearly, equality holds if and only if x_1 = x_2 = = x_n = 1.