Problem
ALG-B2-M04-P014 Maximum and minimum
#14
★★★★☆ Level 4 of 5
Let \(a\le b\le c\) and \(x\le y\le z\). Among all permutations \(p,q,r\) of \(x,y,z\), find the maximum and minimum of \(ap+bq+cr\).
Hint. The maximum occurs in the same order, the minimum in the opposite order.
By rearrangement, the maximum is \(ax+by+cz\), and the minimum is \(az+by+cx\). If some elements among \(a,b,c\) or \(x,y,z\) are equal, several permutations may give the maximum or minimum.
This problem checks full understanding of the extremal version, including equality cases.