Problem
ALG-B2-M04-P010 Fourth powers
#10
★★★☆☆ Level 3 of 5
Prove for \(a,b,c>0\): \[a^4+b^4+c^4\ge a^3b+b^3c+c^3a.\]
Hint. After ordering, compare products \(u_i^3\cdot u_{\sigma(i)}\).
Order \(a,b,c\) as \(u_1\le u_2\le u_3\). Then \(u_1^3\le u_2^3\le u_3^3\). By rearrangement, the largest product sum \(u_i^3\cdot u_{\sigma(i)}\) is \(u_1^4+u_2^4+u_3^4\). The right side is one of these permutations.
Watch that the student does not turn the problem into a long expansion unnecessarily.