Problem
ALG-B2-M04-P008 Four nonnegative numbers
#8
★★★☆☆ Level 3 of 5
Prove for \(a,b,c,d\ge0\): \[4(a^3+b^3+c^3+d^3)\ge(a+b+c+d)(a^2+b^2+c^2+d^2).\]
Hint. Apply Chebyshev to \(a,b,c,d\) and \(a^2,b^2,c^2,d^2\) after ordering.
Order the numbers increasingly. Their squares have the same order. By Chebyshev, \[\frac{\sum a^3}{4}\ge\frac{\sum a}{4}\cdot\frac{\sum a^2}{4}.\] Multiplying by \(16\) gives the inequality.
This generalizes the previous problem and prepares students for \(n\)-variable formulations.