Problem
ALG-B2-M06-P015 Schur degree 4 with two equal variables
#15
★★★★★ Level 5 of 5
Let \(b=c=1\), \(a=t\ge0\). Check \[\sum a^4+abc(a+b+c)\ge\sum_{\mathrm{sym}}a^3b.\]
Hint. Write both sides explicitly and simplify.
The left side is \(t^4+2+t(t+2)=t^4+t^2+2t+2\). The right side is \(2t^3+2t+2\). The difference is \(t^4-2t^3+t^2=t^2(t-1)^2\ge0\).
Shows the typical factor in Schur degree \(4\).