Track / Stage 4 / 297 of 340 #557 of 1964

Problem 557

AMC 12 late, AIME early

Algebra Difficulty 4.9 Prove it

Derive the inequality between the arithmetic mean and the geometric mean from Muirhead's inequality (problem 61424).

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

According to Muirhead's inequality $x_{1}^{n}+\ldots+x_{n}^{n} \geq n x_{1} \ldots x_{n}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.