Problem
ALG-B2-M09-P006 Minimum sum with fixed q
#6
★★★☆☆ Level 3 of 5
Let \(a,b,c\ge0\), \(ab+bc+ca=12\). Prove \(a+b+c\ge6\).
Hint. Use \(p^2\ge3q\).
Let \(p=a+b+c\), \(q=ab+bc+ca=12\). Then \(p^2\ge3q=36\), so \(p\ge6\). Equality occurs at \(a=b=c=2\).
First bound on \(p\) with fixed \(q\).