## [Hints] 2013 APMO Problem 2

2013 APMO 2. Determine all positive integers $n$ for which $\displaystyle \frac{n^2+1}{[\sqrt{n}]^2 + 2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

Hint 1: Let $k := [\sqrt{n}]$. Then $n = k^2 + \alpha$, where $\alpha = 0, 1, \dots 2k$. Think of the problem in terms of $k$ instead.

Hint 2: After rewriting the fraction in the question in terms of $k$ and $\alpha$, perform long division so that the degree of $k$ is lower in the numerator than in the denominator.