Difference between revisions of "2002 AIME II Problems/Problem 15"
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== Problem == | == Problem == | ||
Circles <math>\mathcal{C}_{1}</math> and <math>\mathcal{C}_{2}</math> intersect at two points, one of which is <math>(9,6),</math> and the product of the radii is <math>68.</math> The x-axis and the line <math>y = mx</math>, where <math>m > 0,</math> iare tangent to both circles. It is given that <math>m</math> can be written in the form <math>a\sqrt {b}/c,</math> where <math>a,</math> <math>b,</math> and <math>c</math> are positive integers, <math>b</math> is not divisible by the square of any prime, and <math>a</math> and <math>c</math> are relatively prime. Find <math>a + b + c.</math> | Circles <math>\mathcal{C}_{1}</math> and <math>\mathcal{C}_{2}</math> intersect at two points, one of which is <math>(9,6),</math> and the product of the radii is <math>68.</math> The x-axis and the line <math>y = mx</math>, where <math>m > 0,</math> iare tangent to both circles. It is given that <math>m</math> can be written in the form <math>a\sqrt {b}/c,</math> where <math>a,</math> <math>b,</math> and <math>c</math> are positive integers, <math>b</math> is not divisible by the square of any prime, and <math>a</math> and <math>c</math> are relatively prime. Find <math>a + b + c.</math> | ||
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== Solution == | == Solution == | ||
{{solution}} | {{solution}} | ||
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== See also == | == See also == | ||
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Revision as of 13:32, 19 April 2008
Problem
Circles 
and 
intersect at two points, one of which is 
and the product of the radii is 
The x-axis and the line 
, where 
iare tangent to both circles. It is given that 
can be written in the form 
where 
and 
are positive integers, 
is not divisible by the square of any prime, and 
and are relatively prime. Find 
Solution
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See also
| 2002 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Last Question | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
