2006 AMC 10A Problems/Problem 10

Problem

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

$\mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11$

Solution

Since $\sqrt{x}$ cannot be negative, the outermost radicand is at most 120. We are interested in the number of integer values that the expression can take, so we count the number of squares less than 120, the greatest of which is $10^2=100.$

Thus our set of values is

$\{10^2, 9^2,\ldots,2^2, 1^2, 0^2\}$

And our answer is 11, (E)

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2006 AMC 10A Problems/Problem 10

Problem

Solution

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