1. Determine the functions $f: N^{*} \rightarrow N^{*}$ that satisfy the relation $\frac{f^{2}(n)}{n+f(n)}+\frac{n}{f(n)}=\frac{1+f(n+1)}{2},(\forall) n \geq 1$. Cătălin Cristea, Craiova (GMB 9/2013)
1. Let $f(1)=a \in N^{*} \Rightarrow f(2)=\frac{2\left(a^{3}+a+1\right)}{a^{2}+a}-1$......................................................... $1 p$

We obtain $\frac{4 a+2}{a^{2}+a} \in \mathrm{N}^{*}$.............................................................................................. $1 \mathrm{~s}$

We obtain $\frac{2}{\mathrm{a}+1} \in \mathrm{N}^{*}$, hence $\mathrm{a}=1$.................................................................................

$f(2)=2$............................................................................................................ $1 p$

Assume $\mathrm{f}(\mathrm{k})=\mathrm{k}, \forall 1 \leq \mathrm{k} \leq \mathrm{n}$........................................................................... $1 \mathrm{p}$

From the relation in the statement, we get $f(n+1)=\frac{n^{3}+2 n^{2}}{2 n^{2}} \cdot 2-1=n+1$ thus, by mathematical induction $\mathrm{f}(\mathrm{n})=\mathrm{n}, \forall \mathrm{n} \in \mathrm{N}^{*}$

$.2 \mathrm{p}$