A3. Ordering a multiset

Given a multiset of real numbers {a₁, ...,a_n} find expressions e₁, ...,e_n such that {a₁, ...,a_n} = {e₁, ...,e_n}, and {e_i} is a non-increasing sequence, where each expression is formed from the a_i, the max function, and elementary arithmetic operations.
I will not post the answer to this problem, because I plan to publish it if it is not already known. If it is a known result, I would appreciate a reference.

E16. A Property of rectangles

Someone showed me the following problem at the meeting of ATMIM (Association of Teachers of Mathematics in Massachusetts.) It's proof is a great non-routine use of the Pythagorean theorem. I think the proof is too easy to post, but if anyone can't figure it out, post a comment and I will make the solution available.

Let ABCD be a rectangle, and P any point in the interior. Prove that AP² + PC² = BP² + PD².