Difference between revisions of "2002 AMC 10B Problems/Problem 16"
(New page: 16. For how many integers <math>n</math> is <math>\frac{n}{20-n}</math> the square of an integer? <math>\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\q...) |
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| − | + | == Problem == | |
| + | |||
| + | For how many integers <math>n</math> is <math>\frac{n}{20-n}</math> the square of an integer? | ||
<math>\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 10</math> | <math>\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 10</math> | ||
| + | |||
| + | == Solution == | ||
| + | |||
| + | For <math>n=20</math> the fraction is undefined, for <math>n>20</math> and <math>n<0</math> it is negative, hence not a square. | ||
| + | |||
| + | This leaves <math>0\leq n < 20</math>. | ||
| + | |||
| + | For <math>n=0</math> the fraction equals <math>0</math>, which is a square. | ||
| + | |||
| + | For <math>1\leq n\leq 9</math> the fraction is strictly between <math>0</math> and <math>1</math>. | ||
| + | |||
| + | For <math>n=10</math> the fraction equals <math>1</math>, which is a square. | ||
| + | |||
| + | The next square is <math>4</math>, and this is achieved for <math>n=16</math>, and the square after that is <math>9</math>, achieved for <math>n=18</math>. | ||
| + | |||
| + | That leaves <math>n=19</math>, for which the fraction is <math>19</math>, which is not a square. | ||
| + | |||
| + | In total, there are <math>\boxed{4}</math> squares among these fractions. | ||
| + | |||
| + | == See Also == | ||
| + | |||
| + | {{AMC10 box|year=2002|ab=B|num-b=15|num-a=17}} | ||
Revision as of 06:40, 2 February 2009
Problem
For how many integers 
is 
the square of an integer?
Solution
For 
the fraction is undefined, for 
and 
it is negative, hence not a square.
This leaves 
.
For 
the fraction equals 
, which is a square.
For 
the fraction is strictly between and 
.
For 
the fraction equals , which is a square.
The next square is 
, and this is achieved for 
, and the square after that is 
, achieved for 
.
That leaves 
, for which the fraction is 
, which is not a square.
In total, there are 
squares among these fractions.
See Also
| 2002 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 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 | ||
