5. Compute the smallest positive integer $n$ for which
$$
\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}}
$$
is an integer.
Proposed by: Michael Tang
Answer: 6156
The number $\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}}$ is a positive integer if and only if its square is a perfect square. We have
$$
\begin{aligned}
(\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}})^{2} & =(100+\sqrt{n})+(100-\sqrt{n})+2 \sqrt{(100+\sqrt{n})(100-\sqrt{n})} \\
& =200+2 \sqrt{10000-n} .
\end{aligned}
$$

To minimize $n$, we should maximize the value of this expression, given that it is a perfect square. For this expression to be a perfect square, $\sqrt{10000-n}$ must be an integer. Then $200+2 \sqrt{10000-n}$ is even, and it is less than $200+2 \sqrt{10000}=400=20^{2}$. Therefore, the greatest possible perfect square value of $200+2 \sqrt{10000-n}$ is $18^{2}=324$. Solving
$$
200+2 \sqrt{10000-n}=324
$$
for $n$ gives the answer, $n=6156$.