HMMT-Nov 2018 gen, Problem 5

Compute the smallest positive integer $n$ for which \[ \sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}} \] is an integer.

Solution: 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, $n=6156$. $\fbox{6156}$.

Answer: 6156