Difference between revisions of "2019 AIME I Problems/Problem 15"
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==Problem 15== | ==Problem 15== | ||
| + | Let <math>\overline{AB}</math> be a chord of a circle <math>\omega</math>, and let <math>P</math> be a point on the chord <math>\overline{AB}</math>. Circle <math>\omega_1</math> passes through <math>A</math> and <math>P</math> and is internally tangent to <math>\omega</math>. Circle <math>\omega_2</math> passes through <math>B</math> and <math>P</math> and is internally tangent to <math>\omega</math>. Circles <math>\omega_1</math> and <math>\omega_2</math> intersect at points <math>P</math> and <math>Q</math>. Line <math>PQ</math> intersects <math>\omega</math> at <math>X</math> and <math>Y</math>. Assume that <math>AP=5</math>, <math>PB=3</math>, <math>XY=11</math>, and <math>PQ^2 = \tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
==Solution== | ==Solution== | ||
Revision as of 20:25, 14 March 2019
The 2019 AIME I takes place on March 13, 2019.
Problem 15
Let 
be a chord of a circle 
, and let 
be a point on the chord . Circle 
passes through 
and and is internally tangent to . Circle 
passes through 
and and is internally tangent to . Circles and intersect at points and 
. Line 
intersects at 
and 
. Assume that 
, 
, 
, and 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
See Also
| 2019 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Last Problem | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
