Putnam 1960 A1

Let $n$ be a given positive integer. How many solutions are there in ordered positive integer pairs $(x,y)$ to the equation $\frac{xy}{x+y}=n?$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.