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

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== Problem ==
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#redirect [[2006 AMC 12A Problems/Problem 10]]
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>
 
== Solution ==
 
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
 
 
 
<center><math> \{10^2, 9^2,\ldots,2^2, 1^2, 0^2\} </math></center>
 
 
 
And our answer is <math>11 \Longrightarrow \mathrm{E}</math>.
 
 
 
== See also ==
 
{{AMC10 box|year=2006|ab=A|num-b=9|num-a=11}}
 
 
 
[[Category:Introductory Algebra Problems]]
 

Latest revision as of 23:17, 27 April 2008

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