Problem
ALG-B2-M04-P021 Two powers in a permutation
#21
★★★★★ Level 5 of 5
Prove for \(a,b,c>0\): \[a^6+b^6+c^6\ge a^4b^2+b^4c^2+c^4a^2.\]
Hint. Compare the similarly ordered sequences \(u_i^4\) and \(u_i^2\).
After ordering \(a,b,c\), we have \(u_1^2\le u_2^2\le u_3^2\) and \(u_1^4\le u_2^4\le u_3^4\). By rearrangement, the largest sum \(u_i^4u_{\sigma(i)}^2\) is \(\sum u_i^6\). The right side corresponds to one permutation, so it is not larger than the left side.
Here the student must identify two derived sequences, not just the original variables.