Question 63
Let $n \geq 2$ be an integer. An $n$-tuple $(a_{1}, a_{2}, \ldots, a_{n})$ of positive integers is expensive if there exists a positive integer $k$ such that
$$
(a_{1}+a_{2})(a_{2}+a_{3}) \cdots (a_{n-1}+a_{n})(a_{n}+a_{1})=2^{2k-1}.
$$
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.
$$
(a_{1}+a_{2})(a_{2}+a_{3}) \cdots (a_{n-1}+a_{n})(a_{n}+a_{1})=2^{2k-1}.
$$
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.