Problem
ALG-B2-M04-P020 Fifth power in a cycle
#20
★★★★★ Level 5 of 5
Prove for \(a,b,c>0\): \[a^5+b^5+c^5\ge a^4b+b^4c+c^4a.\]
Hint. This is rearrangement for \(u_i^4\) and \(u_i\).
Order \(a,b,c\) as \(u_1\le u_2\le u_3\). Then \(u_i^4\) are ordered in the same way. The maximum of \(\sum u_i^4u_{\sigma(i)}\) is \(\sum u_i^5\). The right side is one of the permutations, so it is not larger.
This repeats the idea at a higher degree and helps automate method recognition.