Track / Stage 5 / 57 of 400 #657 of 2000

Problem 657

AIME late

Algebra Difficulty 5.1 Prove it

278*. Prove that for any positive values of $x$ and $y$

$x^{5}+y^{5} \geqslant x^{4} y+x y^{4}$

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Instruction. Transform the difference between the right and left sides of the inequality to be proved into the form:

$(x-y)^{2}\left(x^{2}+y^{2}\right)(x+y)$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.