Difference between revisions of "1998 USAMO Problems/Problem 2"
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| − | + | == Problem == | |
| + | Let <math>{\cal C}_1</math> and <math>{\cal C}_2</math> be concentric circles, with <math>{\cal C}_2</math> in the interior of <math>{\cal C}_1</math>. From a point <math>A</math> on <math>{\cal C}_1</math> one draws the tangent <math>AB</math> to <math>{\cal C}_2</math> (<math>B\in {\cal C}_2</math>). Let <math>C</math> be the second point of intersection of <math>AB</math> and <math>{\cal C}_1</math>, and let <math>D</math> be the midpoint of <math>AB</math>. A line passing through <math>A</math> intersects <math>{\cal C}_2</math> at <math>E</math> and <math>F</math> in such a way that the perpendicular bisectors of <math>DE</math> and <math>CF</math> intersect at a point <math>M</math> on <math>AB</math>. Find, with proof, the ratio <math>AM/MC</math>. | ||
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| + | == Solution == | ||
| + | {{solution}} | ||
| + | |||
| + | == See Also == | ||
| + | {{USAMO newbox|year=1998|num-b=1|num-a=3}} | ||
| + | |||
| + | [[Category:Olympiad Geometry Problems]] | ||
Revision as of 12:58, 16 April 2012
Problem
Let 
and 
be concentric circles, with in the interior of . From a point 
on one draws the tangent 
to (
). Let 
be the second point of intersection of and , and let 
be the midpoint of . A line passing through intersects at 
and 
in such a way that the perpendicular bisectors of 
and 
intersect at a point 
on . Find, with proof, the ratio 
.
Solution
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See Also
| 1998 USAMO (Problems • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAMO Problems and Solutions | ||
