Problem
ALG-B2-M11-P019 Fifth-degree Schur
#19
★★★★★ 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 in the form \(\sum a^3(a-b)(a-c)\ge0\).
By Schur, \[\sum a^3(a-b)(a-c)\ge0.\] Expanding gives \[\sum a^5+abc(a^2+b^2+c^2)-\sum_{\mathrm{sym}}a^4b\ge0,\] which is equivalent to the required inequality.
Teaching goal: the student should first recognise the type of estimate, then choose the tool.