Problem
ALG-B2-M10-P008 Fixed pairwise sum
#8
★★★★★ Level 5 of 5
Let \(a,b,c>0\), \(ab+bc+ca=3\). Prove \(abc\le1\).
Hint. Use \(p^2\ge3q\) and \(q^2\ge3pr\).
Let \(p=a+b+c\), \(q=3\), \(r=abc\). From \(p^2\ge3q\), \(p\ge3\). From \(q^2\ge3pr\), we get \(9\ge3pr\), so \(r\le3/p\le1\).
Teaching goal: choose a combination of methods and always check the equality case.