KöMaL 2017 B4876

Find the largest positive integer $n$ for which there exists exactly one integer $k$ such that $\frac{10}{11}<\frac{n}{k+n}<\frac{11}{12}?$

Official text KöMaL. 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.