<s>[INST] Find all possible integers $n$ such that $\sqrt{n + 2} + \sqrt{n + \sqrt{n + 2}}$ is an integer. 
Find all possible integers $n$ such that $m = \sqrt{n + 2} + \sqrt{n + \sqrt{n + 2}}$ is an integer.

Guess what? This problem is adapted from a recent competition. There have been a solution below for you to check out. I am aware of the fact that there are other solutions that are more practical and suitable in test setting.
 [/INST] We have that $n+2 = a^2, n + a = b^2$, where $a$ and $b$ are non-negative integers.
If $a > 2$, then $$a^2  = n+2 < n+a = a^2 + a - 2 < a^2 + 2a + 1.$$ 
This means that $n+a$ cannot be a perfect square as it's bounded between 2 consecutive perfect squares, which is a contradiction. Hence $ 0 \leq a \leq 2$.
If $ a= 0$, then $n = -2$ but $n+a = -2$ is not a perfect square. No solution.
If $a = 1$, then $n = - 1$ and $n+a = 0$ is a pefect square. This gives $m=1$.
If $a=2$, then $n=2$ and $n+a = 4$ is a perfect square. This gives $m=4$.
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