HMMT February

Suppose $f(f(n))=n$, and let $m=f(n)$. Note that we have $mn \mid m^{2}+n^{2}+1$. First we find all pairs of positive integers that satisfy this condition, using Vieta root jumping. Suppose $m^{2}+n^{2}+1=kmn$, for some positive integer $k$. Considering this as a quadratic in $m$, let the other root (besides $m$) be $m^{\prime}$. We have $m^{\prime}+m=kn$, so $m^{\prime}$ is an integer. Also, $mm^{\prime}=n^{2}+1$. So if $m>n$ then $m^{\prime} \leq n$. So if we have a solution $(m, n)$ we can find a smaller solution $\left(n, m^{\prime}\right)$. In particular, it suffices to find all small solutions to describe all solutions. A minimal solution must have $m=n$, which gives only $m=n=1$. We have that $k=3$. Now the recurrence $a_{0}=a_{1}=1, a_{n}+a_{n+2}=3a_{n+1}$ describes all solutions with consecutive terms. In fact this recurrence gives precisely other Fibonacci number: $1,1,2,5,13,34,89,233, \ldots$ Checking these terms gives an answer of 89.