Problem
ALG-B2-M11-P005 Product of pairwise sums
#5
★★★☆☆ Level 3 of 5
Prove for \(a,b,c>0\): \[(a+b)(b+c)(c+a)\ge8abc.\]
Hint. Use \(a+b\ge2\sqrt{ab}\) for each pair.
We have \(a+b\ge2\sqrt{ab}\), \(b+c\ge2\sqrt{bc}\), \(c+a\ge2\sqrt{ca}\). Multiplying gives \((a+b)(b+c)(c+a)\ge8abc\).
Teaching goal: the student should first recognise the type of estimate, then choose the tool.