Problem
ALG-B2-M02-P003 Sum equals 15
#3
★★☆☆☆ Level 2 of 5
Let \(a,b,c>0\) and \(a+b+c=15\). Prove \(abc\le125\).
Hint 1. Estimate \(\sqrt[3]{abc}\).
Hint 2. Equality is possible at \(a=b=c=5\).
By AM-GM, \(\sqrt[3]{abc}\le\frac{a+b+c}{3}=5\). Cubing gives \(abc\le125\).
AM-GM module training problem. Method tags: am-gm, fixed-sum.