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Problem 1174

AIME late

Algebra Difficulty 5.1 Prove it Ukrainian National Mathematical Olympiad, Third Round, First Tour · Ukraine

You are given $n \ge 4$ positive real numbers. It turned out that their $\frac{n(n-1)}{2}$ pairwise products form an arithmetic progression in some order. Prove that all of these numbers are equal.

(Anton Trygub)

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

If some two products are equal, then all products are equal, and all numbers are equal. If some two numbers are equal, then some products are equal, so all numbers are equal. Now consider 4 largest numbers $a < b < c < d$ . The largest two products are $cd, bd$ . Then the difference of the progression is $cd - bd$ . But then $ac - ab = a(c - b) < d(c - b)$ , so the difference between some two elements of the progression is smaller than the difference of the progression, which is impossible, a contradiction.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.