Problem
ALG-B2-M10-P010 Sum of fourth powers
#10
★★★★★ Level 5 of 5
Prove for \(a,b,c\ge0\): \[a^4+b^4+c^4+a b c(a+b+c)\ge a^3b+b^3c+c^3a+a^3c+b^3a+c^3b.\]
Hint. This is the same Schur degree \(4\), but the right side is written explicitly.
The right side is \(\sum_{\mathrm{sym}}a^3b\). Thus the statement is Schur degree \(4\), proved via UVW: it is enough to check \(b=c\) and the boundary, where nonnegative squared factors appear.
Teaching goal: choose a combination of methods and always check the equality case.