2006 AMC 10A Problems/Problem 10

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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 $11 \Longrightarrow \mathrm{E}$.

See also

2006 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions
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