Problem
ALG-B2-M12-P004 Set 1. Fifth-degree Schur
#4
★★★★★ Level 5 of 5
Prove for \(a,b,c\ge0\): \[\sum a^5+abc(a^2+b^2+c^2)\ge\sum_{\mathrm{sym}}a^4b.\]
Hint. Use Schur: \(\sum a^3(a-b)(a-c)\ge0\).
By Schur, \(\sum a^3(a-b)(a-c)\ge0\). Expanding gives exactly \(\sum a^5+abc\sum a^2-\sum_{\mathrm{sym}}a^4b\ge0\).
Mock set 1: the problem is intended for independent method selection without an explicit cue in the statement.