Problem
ALG-B1-M12-P027 Variant 7, Problem 3
#27
★★★★★ Level 5 of 5
Prove for \(a,b,c\ge0\): \(a^3+b^3+c^3+3abc\ge\sum_{\mathrm{sym}}a^2b\).
This is Schur's inequality.
Let \(a\ge b\ge c\). The difference equals \(\sum a(a-b)(a-c)=(a-b)^2(a+b-c)+c(a-c)(b-c)\ge0\).
Mock olympiad problem. Tags: schur.