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Understand the problem

Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$for all positive integers $a,b$.

APMO 2019

Number Theory
Medium
Suggested Book
An Excursion in Mathematics

Do you really need a hint? Try it first!

Considering prime numbers might be a good idea.
Show that, for any prime $p$, $f(p)|p^2$. Hence or otherwise show that $f(p)=p$.
If $gcd(p,b)=1$ for a prime $p$, then show that $p+b|p+f(b)$. This is equivalent to $p+b|f(b)-b$. Note that the RHS is independent of $p$.
As in $p+b|f(b)-b$ the RHS is fixed and the LHS is arbitrarily large, the RHS has to be zero. Hence $f(b)=b$ for every natural number $b$.

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