Ukrainian National Mathematical Olympiad

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