1. Determine the functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ that satisfy the relation $\frac{f^{2}(n)}{n+f(n)}+\frac{n}{f(n)}=\frac{1+f(n+1)}{2}$, for any $n \in \mathbb{N}^{*}$.

G.M. $9 / 2013$
Solution. Let $f(1)=k \in \mathbb{N}^{*}$ and for $n=1$ the relation from the statement becomes $\frac{k^{2}}{k+1}+\frac{1}{k}=\frac{1+f(2)}{2} \Leftrightarrow$ $\frac{2 k^{3}+2 k+2}{k^{2}+k}=1+f(2) \Leftrightarrow 2 k-2+\frac{4 k+2}{k^{2}+k}=1+f(2) \Rightarrow k^{2}+k \mid 4 k+2 \Rightarrow k^{2}+k \mid 4 k^{2}+2 k$ and $k^{2}+k \mid 4 k^{2}+4 k$, from which it follows that $k^{2}+k \mid 2 k$ and necessarily $k=1$. Therefore, $f(1)=1$.

We inductively assume that $f(n)=n$ and using the relation from the hypothesis we obtain $f(n+1)=n+1$.

In conclusion, the only function with the given property is $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}, f(n)=n, \forall n \geq 1$.

Grading.

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