One-Line Proof of the AM-GM Inequality

Abstract

x1x2 xn n p (all xi [ 0). This is perhaps the best known and most useful nontrivial inequality in mathematics, with a large number of interesting proofs in the literature. A very clever proof of this inequality, which appeared of late in [1], has given me motivation to communicate a very short proof of it, which I used to present to Iran’s Olympiad team, preparing for the IMO, several years ago. Let me, before going any further, give a generalization of the AM-GM inequality in the case n = 2 . It is trivial to see that, whenever 0\a b and x [ 0; a x b if and only if a þ b x þ ab x . The latter inequality is stronger than the AMGM inequality for n = 2 (indeed, just choose x 1⁄4 ffiffiffiffiffi