Problem
ALG-B2-M04-P004 Chebyshev for three terms
#4
★★☆☆☆ Level 2 of 5
Let \(a\le b\le c\) and \(x\le y\le z\). Prove \[3(ax+by+cz)\ge(a+b+c)(x+y+z).\]
Hint. Use the formula with pairwise differences.
Expanding the difference gives \[3\sum ax-\sum a\sum x=(b-a)(y-x)+(c-a)(z-x)+(c-b)(z-y).\] Every term on the right is nonnegative because both sequences are increasing.
This is the core form that will later be recognized in power-sum problems.