Problem
ALG-B2-M01-P008 Zero sum
#8
★★★★☆ Level 4 of 5
Let \(a+b+c=0\). Prove that \(ab+bc+ca\le0\). When can equality occur?
Hint 1. Square \(a+b+c\).
Hint 2. Use \(0=(a+b+c)^2\).
Since \(a+b+c=0\), \[0=(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca).\] Therefore \[ab+bc+ca=-\frac{a^2+b^2+c^2}{2}\le0.\] Equality is possible only when \(a=b=c=0\).
Module training problem. Method tags: squares, normalisation, equality-case.