Problem
ALG-B1-M07-P024 First Schur inequality
#24
★★★★★ Level 5 of 5
Let \(a,b,c\ge0\). Prove \[a^3+b^3+c^3+3abc\ge a^2b+a^2c+b^2a+b^2c+c^2a+c^2b.\]
Assume without loss of generality that \(a\ge b\ge c\), and consider \(\sum a(a-b)(a-c)\).
The difference between the left and right sides is \(\sum a(a-b)(a-c)\). Assume \(a\ge b\ge c\). Then \[\sum a(a-b)(a-c)=(a-b)^2(a+b-c)+c(a-c)(b-c).\]
Both terms are nonnegative: \(a-b\ge0\), \(a+b-c\ge0\), \(c\ge0\), \(a-c\ge0\), \(b-c\ge0\). Hence the whole sum is nonnegative, and the inequality is proved.
This is a challenge problem: it shows that sorting variables can turn a symmetric inequality into an obvious one.