Problem
ALG-B2-M04-P009 Cyclic cubes
#9
★★★☆☆ Level 3 of 5
Prove for \(a,b,c>0\): \[a^3+b^3+c^3\ge a^2b+b^2c+c^2a.\]
Hint. Consider the ordered sequences \(u_i^2\) and \(u_i\).
Let \(u_1\le u_2\le u_3\) be \(a,b,c\) in increasing order. Then \(u_1^2\le u_2^2\le u_3^2\). By rearrangement, the largest sum \(\sum u_i^2u_{\sigma(i)}\) is \(\sum u_i^3\). The right side of the original inequality is one such permutation, so it is not larger.
A good exercise in translating cyclic notation into permutation language.