Problem
ALG-B2-M11-P004 Two sums with fixed product
#4
★★★☆☆ Level 3 of 5
Let \(a,b,c>0\) and \(abc=1\). Prove \[a+b+c+ab+bc+ca\ge6.\]
Hint. Estimate \(a+b+c\) and \(ab+bc+ca\) separately.
From \(abc=1\), AM-GM gives \(a+b+c\ge3\). Also \(ab\cdot bc\cdot ca=(abc)^2=1\), so \(ab+bc+ca\ge3\). Adding gives \(6\).
Teaching goal: the student should first recognise the type of estimate, then choose the tool.