solution: 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.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  For how many real values of $x$ is $\sqrt{120-\sqrt{x}}$ an integer?
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 11
extra_info.question: For how many real values of $x$ is $\sqrt{120-\sqrt{x}}$ an integer?
extra_info.split: test