Functional equation with function defined on $\mathbb{N}^{*}$

Question

Let $ f:\mathbb{N}^{*} \mapsto \mathbb{N}^{*} $ be a function with the following property: $$ \frac{f(x+1)f(x)-2x}{f(x)}=\frac{2f^2(x)}{x+f(x)}-1$$ Determine all functions with this property. (I'm sure the only function is $f(x)=x$).

Answer 1

Let $x=1$ and multiply both sides by $f(1)$ to determine that $\displaystyle \frac{2f^3(1)}{1+f(1)}$ is an integer since all other terms are integers. Since $1+f(1)$ and $f(1)$ are relatively prime, it follows that $(1+f(1))|2$, and since $f(1)$ is positive, $f(1) = 1$.

By induction all values of $f$ are determined by the value of $f(1)$ (just solve for $f(x+1)$ in terms of $f(x)$) so there is at most one such function. Since $f(x) = x$ works, it is the only such function.