Olympiad Maths Prep

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Algebra Difficulty 5.1 AIME, harder Prove it Ukraine

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

(Anton Trygub)

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<da < b < c < d. The largest two products are cd,bdcd, bd. Then the difference of the progression is cd−bdcd - bd. But then ac−ab=a(c−b)<d(c−b)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.

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