Problem
ALG-B2-M04-P003 Two adjacent swaps
#3
★★☆☆☆ Level 2 of 5
Let \(a\le b\le c\) and \(x\le y\le z\). Prove \[ay+bz+cx\le ax+by+cz.\]
Hint. First swap \(z\) and \(x\) at the coefficients \(b,c\), then swap \(y\) and \(x\) at the coefficients \(a,b\).
By the one-swap inequality, \(bz+cx\le bx+cz\), since \(b\le c\) and \(x\le z\). Also \(ay+bx\le ax+by\), since \(a\le b\) and \(x\le y\). Adding gives the result.
The goal is to show how local swaps build a proof for a permutation.