Problem
ALG-B2-M07-P011 Product with fixed sum
#11
★★★★☆ Level 4 of 5
Let \(a,b,c\ge0\), \(a+b+c=1\). Prove \[abc\le\frac1{27}.\]
Hint. This is AM-GM, but write the result in homogeneous form.
By AM-GM, \(\frac{a+b+c}{3}\ge\sqrt[3]{abc}\). Since the sum is \(1\), \(\sqrt[3]{abc}\le1/3\), hence \(abc\le1/27\). Homogeneous form: \(abc\le\frac{(a+b+c)^3}{27}\).
The key point is seeing how the constant \(\frac1{27}\) is homogenized.