B2. Find all natural numbers $n$ and prime numbers $p$ for which $\sqrt{n+\frac{p}{n}}$ is a natural number.
B2. Let $\sqrt{n+\frac{p}{n}}=k$, where $k$ is a natural number, so $n+\frac{p}{n}=k^{2}$. From this, it follows that $n$ divides $p$. Since $p$ is a prime number, $n$ must be $1$ or $n=p$. In both cases, we get the equation $1+p=k^{2}$ or $p=(k+1)(k-1)$. From this, it follows that $k=2$, so $p=3$. We obtain the solutions $n=1, p=3$ and $n=3, p=3$.
Conclusion that $n$ divides $p$ ..... 2 points
Observation that $n=1$ or $n=p$ ..... 1 point
Writing the equation $p=(k-1)(k+1)$ ..... 1 point
Conclusion that $k=2$ or $p=3$ ..... 1 point
Writing both solutions ..... 1 point
(If the contestant only writes both solutions, award 1 point.)