|
|
| Line 1: |
Line 1: |
| − | == Problem ==
| + | #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]]
| |
