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Determine all solutions, equation

Source: Austrian Mathematics Olympiad Regional Competition 2020, problem 1

July 23, 2020
algebraAustria

Problem Statement

Let aa be a positive integer. Determine all aa such that the equation (1+1x)⋅(1+1x+1)⋯(1+1x+a)=a−x \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 xx. For every such aa state the respective solutions.
(Richard Henner)
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