59th Ukrainian National Mathematical Olympiad

Question

Some natural numbers are placed around a circle in such a way that the product of any two neighboring numbers is a perfect square. Prove that the product of any (not necessarily neighboring) two numbers is also a perfect square. (Arseniy Nikolaev)

Accepted Answer

Let us denote numbers as a_1, a_2, , a_k. From the problem statement it follows that for any 1 i < j k the product (a_i a_i+1)(a_i+1 a_i+2) (a_j-1 a_j) = n^2 is a perfect square. Hence the product a_i a_j = n^2a_i+1^2 a_i+2^2 a_j-1^2 is also a perfect square, as the natural ratio of two perfect squares, Q.E.D.