Problem
ALG-B2-M10-P007 Product of pairwise sums
#7
★★★★★ Level 5 of 5
Let \(a,b,c\ge0\), \(a+b+c=3\). Prove \[(a+b)(b+c)(c+a)\le8.\]
Hint. Replace \(a+b\) by \(3-c\).
We have \((a+b)(b+c)(c+a)=(3-c)(3-a)(3-b)\). The sum of the factors is \(6\). By AM-GM, the product is at most \((6/3)^3=8\). Equality holds at \(a=b=c=1\).
Teaching goal: choose a combination of methods and always check the equality case.