Problem
ALG-B2-M06-P009 Schur with two equal variables
#9
★★★☆☆ Level 3 of 5
Let \(b=c=1\), \(a=t\ge0\). Check the inequality \[\sum a^3+3abc\ge\sum_{\mathrm{sym}}a^2b.\]
Hint. Substitute \(a=t\), \(b=c=1\), and factor the difference.
The left side is \(t^3+2+3t\). The right side is \(2t^2+2t+2\). The difference is \(t^3-2t^2+t=t(t-1)^2\ge0\).
Trains the typical \(b=c\) check.