Problem
ALG-B2-M07-P005 Homogenizing a constant
#5
★★☆☆☆ Level 2 of 5
Let \(a,b,c\ge0\), \(a+b+c=1\). Prove \[ab+bc+ca\le\frac13.\]
Hint. Replace \(\frac13\) by \(\frac{(a+b+c)^2}{3}\).
Since \(a+b+c=1\), it is enough to prove \(ab+bc+ca\le\frac{(a+b+c)^2}{3}\), i.e. \(3q\le p^2\). This follows from \(p^2-3q=\frac12\sum(a-b)^2\ge0\).
Teaches how to turn a numerical constant into a homogeneous expression.