Problem
ALG-B2-M06-P020 Schur degree 4 with fixed sum
#20
★★★★★ Level 5 of 5
Let \(a,b,c\ge0\) and \(a+b+c=3\). Prove \[a^4+b^4+c^4+3abc\ge\sum_{\mathrm{sym}}a^3b.\]
Hint. This is Schur degree \(4\) with \(p=3\).
The general Schur degree \(4\) is \(\sum a^4+abc(a+b+c)\ge\sum_{\mathrm{sym}}a^3b\). Under \(a+b+c=3\), we have \(abc(a+b+c)=3abc\). Thus the required inequality follows directly from Schur degree \(4\).
Final module problem: recognize the general form inside a normalized condition.