Problem
ALG-B2-M07-P019 Homogenize the problem
#19
★★★★★ Level 5 of 5
Let \(a,b,c\ge0\), \(a+b+c=1\). Prove \[a^2+b^2+c^2+ab+bc+ca\ge\frac23.\] Then write the homogeneous version of this inequality without the condition \(a+b+c=1\).
Hint. Use \(\sum a^2\ge q\) and \(p=1\).
Let \(q=ab+bc+ca\). Since \(p^2=\sum a^2+2q=1\), we have \(\sum a^2+q=1-q\). Also \(q\le1/3\), so \(1-q\ge2/3\). The homogeneous version is \[\sum a^2+ab+bc+ca\ge\frac{2}{3}(a+b+c)^2.\]
This problem deliberately forces the student to homogenize the result.