Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 14"
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+ | ==Problem== | ||
+ | Circles <math>\omega_1</math> and <math>\omega_2</math> are centered on opposite sides of line <math>l</math>, and are both tangent to <math>l</math> at <math>P</math>. <math>\omega_3</math> passes through <math>P</math>, intersecting <math>l</math> again at <math>Q</math>. Let <math>A</math> and <math>B</math> be the intersections of <math>\omega_1</math> and <math>\omega_3</math>, and <math>\omega_2</math> and <math>\omega_3</math> respectively. <math>AP</math> and <math>BP</math> are extended past <math>P</math> and intersect <math>\omega_2</math> and <math>\omega_1</math> at <math>C</math> and <math>D</math> respectively. If <math>AD = 3, AP = 6, DP = 4,</math> and <math>PQ = 32</math>, then the area of triangle <math>PBC</math> can be expressed as <math>\frac{p\sqrt{q}}{r}</math>, where <math>p, q,</math> and <math>r</math> are positive integers such that <math>p</math> and <math>r</math> are coprime and <math>q</math> is not divisible by the square of any prime. Determine <math>p + q + r</math>. | ||
− | + | ==Solution== | |
+ | Invert about a circle with radius 1 and center P. Note that since all relevant circles and lines go through P, they all are transformed into lines, and <math>\omega_1,\omega_2, l</math> are all tangent at infinity (i.e. parallel). That was the crux move; some more basic length chasing using similar triangles gets you the answer. | ||
+ | {{solution}} | ||
+ | |||
+ | ==See Also== | ||
+ | {{Mock AIME box|year=Pre 2005|n=3|num-b=13|num-a=15}} |
Revision as of 10:37, 4 April 2012
Problem
Circles and are centered on opposite sides of line , and are both tangent to at . passes through , intersecting again at . Let and be the intersections of and , and and respectively. and are extended past and intersect and at and respectively. If and , then the area of triangle can be expressed as , where and are positive integers such that and are coprime and is not divisible by the square of any prime. Determine .
Solution
Invert about a circle with radius 1 and center P. Note that since all relevant circles and lines go through P, they all are transformed into lines, and are all tangent at infinity (i.e. parallel). That was the crux move; some more basic length chasing using similar triangles gets you the answer. This problem needs a solution. If you have a solution for it, please help us out by adding it.
See Also
Mock AIME 3 Pre 2005 (Problems, Source) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |