Problem
ALG-B2-M06-P007 Maximum of q for fixed p
#7
★★★☆☆ Level 3 of 5
Let \(a,b,c\ge0\) and \(a+b+c=p\). Prove \[ab+bc+ca\le\frac{p^2}{3}.\]
Hint. Use \(p^2\ge3q\).
By \(p^2-3q=\frac12\sum(a-b)^2\ge0\), we get \(q\le\frac{p^2}{3}\). Equality is possible only at \(a=b=c\).
Fixing \(p\) and controlling \(q\) is a common first UVW step.