EGMO 2017 Problem 5

Let $n \geq 2$ be an integer. An $n$-tuple $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of positive integers is expensive if there exists a positive integer $k$ such that

$$\left(a_{1}+a_{2}\right)\left(a_{2}+a_{3}\right) \cdots \cdots\left(a_{n-1}+a_{n}\right)\left(a_{n}+a_{1}\right)=2^{2 k-1} .$$

a) Find all positive integers $n \geq 2$ for which there exists an expensive $n$-tuple. b) Prove that for every positive integer $m$ there exists an integer $n \geq 2$ such that $m$ belongs to an expensive $n$-tuple.

There are exactly $n$ factors in the product on the left hand side. Harun Hindija, Bosnia and Herzegovina