AT-OEMO-REG 2020 Problem 1

Let $a$ be a positive integer. Determine all $a$ such that the equation $$\biggl( 1+\frac{1}{x} \biggr) \cdot \biggl( 1+\frac{1}{x+1} \biggr) \cdots \biggl( 1+\frac{1}{x+a} \biggr)=a-x$$ has at least one integer solution for $x$ . For every such $a$ state the respective solutions.

(Richard Henner)