I hope I'm not making stupid mistakes If that big radical equals $q \in \mathbb Q$, then we square repeatedly to deduce that $q$ is the root of a monic integer polynomial, so it's an integer. In particular, all the "smaller" square roots are also integer. Now, going from $\sqrt{\sigma (n)}$ "upwards", we see that each square root is $\leq k+1$, where $k = \left\lfloor \sqrt n \right\rfloor$. Now, the number $(k-1)^{2}+1$ can't be $\sigma (n)$ for $k \geq 2$, so it's $\sigma (i)$ for some $i$. We know that $\sqrt{\sigma (i)+\sqrt{\ldots}}$ is an integer and that $\sqrt{\ldots}\in \overline{1,k+1}$. But the number we obtain is clearly between two consecutive perfect squares, so that's it (for $k \geq 4$). Now suppose that $k \leq 3$. If $n$ isn't a perfect square, then $k^{2}+1 \leq n$ and by the same reasoning as above we obtain a perfect square strictly between two consecutive perfect squares (for $k \geq 2$). Only the cases $n=1,2,3,4,9$ remain. If $n = 9$, then we get a contradiction choosing $5$ in the previous reasoning. From here it's obvious that $\boxed{n = 1,3}$.

Why each square root is $\leq k+1$ ? For example, $\sqrt{8+\sqrt{7}}> \lfloor \sqrt{8}\rfloor+1$

Remember that each square root $\sqrt{\sigma(i)+\sqrt{\ldots}}$ must be integer! Now try to prove my statement by finite induction, from $i=n$ to $i=1$. If the details don't work out, please tell me.

In more details will be better

Euler_bg wrote: In more details will be better You can download the official solution for all problems from the BMO 2007 web page http://www.hms.gr/bmo2007/main.html Alexandros

Thank very much, cretanman!

cretanman wrote: Euler_bg wrote: In more details will be better You can download the official solution for all problems from the BMO 2007 web page http://www.hms.gr/bmo2007/main.html Alexandros Link was dead. Who have this file?

okay sorry this is a really old post but can anyone explain what σ(n) signifies? thanks a lot

It is just the image of n by $\sigma$ see 3 for an example https://www.math.umd.edu/~immortal/MATH403/lecturenotes/ch5.pdf