Track / Stage 6 / 20 of 400 #1020 of 2000

Problem 1020

National olympiad, first round

Algebra Difficulty 6.0 Prove it

11. If $a, b, c$ are non-negative numbers, then
$\sqrt{\frac{a^{3}}{a^{3}+(b+c)^{3}}}+\sqrt{\frac{b^{3}}{b^{3}+(c+a)^{3}}}+\sqrt{\frac{c^{3}}{c^{3}+(a+b)^{3}}} \geqslant 1$

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

11. (2007.01.17) Brief Proof: See "Chapter 3. Proving Inequalities by the Method of Amplification" Example 6.

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