Difference between revisions of "2006 AMC 10A Problems/Problem 10"

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== Problem ==
 
== Problem ==
For how many real values of <math>\displaystyle x</math> is <math>\sqrt{120-\sqrt{x}}</math> an integer?
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For how many real values of <math>\displaystyle x</math> is <math>\sqrt{120-\sqrt{x}}</math> an [[integer]]?
  
 
<math> \mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11 </math>
 
<math> \mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11 </math>
 
== Solution ==
 
== Solution ==
Since <math>\sqrt{x}</math> 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 <math>10^2=100.</math>
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Since <math>\sqrt{x}</math> cannot be [[negative]], the outermost [[radicand]] is at most <math>120</math>. We are interested in the number of integer values that the expression can take, so we count the number of squares less than <math>120</math>, the greatest of which is <math>10^2=100</math>.
  
 
Thus our set of values is  
 
Thus our set of values is  
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<center><math> \{10^2, 9^2,\ldots,2^2, 1^2, 0^2\} </math></center>
 
<center><math> \{10^2, 9^2,\ldots,2^2, 1^2, 0^2\} </math></center>
  
And our answer is '''11, (E)'''
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And our answer is <math>11 \Longrightarrow \mathrm{E}</math>.
  
== See Also ==
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== See also ==
*[[2006 AMC 10A Problems]]
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{{AMC10 box|year=2006|ab=A|num-b=9|num-a=11}}
 
 
*[[2006 AMC 10A Problems/Problem 9|Previous Problem]]
 
 
 
*[[2006 AMC 10A Problems/Problem 11|Next Problem]]
 
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]

Revision as of 16:12, 28 February 2007

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
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All AMC 10 Problems and Solutions
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