Suppose that  $k = \sqrt{120 - \sqrt{x}}$ is an integer. Then $0\le k \le \sqrt{120}$, and because $k$ is an integer, we have $0\le k \le 10$. Thus there are 11 possible integer values of $k$. For each such $k$, the corresponding value of $x$ is $\left(120 - k^2\right)^2$. Because  $\left(120 -
k^2\right)^2$ is positive and decreasing for $0\le k \le 10$, the $\boxed{11}$ values of $x$ are distinct.
Question:
For how many real values of $x$ is $\sqrt{120-\sqrt{x}}$ an integer?
Answer:
Let's think step by step.

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11
For how many real values of $x$ is $\sqrt{120-\sqrt{x}}$ an integer?
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