Problem
ALG-B2-M09-P004 Maximum pairwise sum
#4
★★☆☆☆ Level 2 of 5
Let \(a,b,c\ge0\), \(a+b+c=1\). Prove \(ab+bc+ca\le\frac13\).
Hint. Use \((a+b+c)^2\ge3(ab+bc+ca)\).
Since \((a+b+c)^2\ge3(ab+bc+ca)\), with \(a+b+c=1\) we get \(1\ge3(ab+bc+ca)\). Equality holds at \(a=b=c=1/3\).
Shows how a fixed sum controls \(q\).