2006 AMC 12A Problem 10

For how many real values of

x

\sqrt{120 - \sqrt{x}}

an integer?

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Solution

Let

k = \sqrt{120 - \sqrt{x}}

be an integer. Then

k \ge 0

and

k^2 = 120 - \sqrt{x} \le 120,

0 \le k \le 10.

Each such

k

gives

\sqrt{x} = 120 - k^2 \ge 0,

hence a distinct value

x = (120 - k^2)^2.

That is

11

values. Thus, the correct answer is E.

Concepts: radical · counting integers in a range

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