Difference between revisions of "2024 AIME I Problems/Problem 10"
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==Problem== | ==Problem== | ||
Let <math>ABC</math> be a triangle inscribed in circle <math>\omega</math>. Let the tangents to <math>\omega</math> at <math>B</math> and <math>C</math> intersect at point <math>D</math>, and let <math>\overline{AD}</math> intersect <math>\omega</math> at <math>P</math>. If <math>AB=5</math>, <math>BC=9</math>, and <math>AC=10</math>, <math>AP</math> can be written as the form <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime integers. Find <math>m + n</math>. | Let <math>ABC</math> be a triangle inscribed in circle <math>\omega</math>. Let the tangents to <math>\omega</math> at <math>B</math> and <math>C</math> intersect at point <math>D</math>, and let <math>\overline{AD}</math> intersect <math>\omega</math> at <math>P</math>. If <math>AB=5</math>, <math>BC=9</math>, and <math>AC=10</math>, <math>AP</math> can be written as the form <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime integers. Find <math>m + n</math>. | ||
+ | |||
+ | ==Diagram== | ||
+ | <asy> | ||
+ | import olympiad; | ||
+ | |||
+ | unitsize(15); | ||
+ | |||
+ | pair A, B, C, D, E, F, P, O; | ||
+ | |||
+ | C = origin; A = (10,0); B = (7.8, 4.4899); | ||
+ | draw(A--B--C--cycle); draw(A..B..C..cycle, red+dotted); | ||
+ | |||
+ | O = circumcenter(A, B, C); | ||
+ | |||
+ | E = rotate(90,B) * (O); | ||
+ | F = rotate(90,C) * (O); | ||
+ | |||
+ | D = IP(B..E + (B-E)*4, C..F + (C-F)*-3); | ||
+ | |||
+ | draw(B--D--C--D--A); | ||
+ | |||
+ | P = IP(D..A, A..B..C); | ||
+ | |||
+ | dot(A); dot(B); dot(C); dot(D); dot(P); | ||
+ | label("$A$", A, dir(335)); | ||
+ | label("$B$", B, dir(65)); | ||
+ | label("$C$", C, dir(200)); | ||
+ | label("$D$", D, dir(135)); | ||
+ | label("$P$", P, dir(235)); | ||
+ | </asy> | ||
==Solution 1== | ==Solution 1== |
Revision as of 18:47, 18 August 2024
Contents
[hide]Problem
Let be a triangle inscribed in circle . Let the tangents to at and intersect at point , and let intersect at . If , , and , can be written as the form , where and are relatively prime integers. Find .
Diagram
Solution 1
From the tangency condition we have . With LoC we have and . Then, . Using LoC we can find : . Thus, . By Power of a Point, so which gives . Finally, we have .
~angie.
Solution 2
We know is the symmedian, which implies where is the midpoint of . By Appolonius theorem, . Thus, we have
~Bluesoul
Solution 3
Extend sides and to points and , respectively, such that and are the feet of the altitudes in . Denote the feet of the altitude from to as , and let denote the orthocenter of . Call the midpoint of segment . By the Three Tangents Lemma, we have that and are both tangents to , and since is the midpoint of , . Additionally, by angle chasing, we get that: Also, Furthermore, From this, we see that with a scale factor of . By the Law of Cosines, Thus, we can find that the side lengths of are . Then, by Stewart's theorem, . By Power of a Point, Thus, Therefore, the answer is .
~mathwiz_1207
Solution 4 (LoC spam)
Connect lines and . From the angle by tanget formula, we have . Therefore by AA similarity, . Let . Using ratios, we have Similarly, using angle by tangent, we have , and by AA similarity, . By ratios, we have However, because , we have so Now using Law of Cosines on in triangle , we have Solving, we find . Now we can solve for . Using Law of Cosines on we have
Solving, we get Now we have a system of equations using Law of Cosines on and ,
Solving, we find , so our desired answer is .
~evanhliu2009
Solution 5
Following from the law of cosines, we can easily get , , .
Hence, , , . Thus, .
Denote by the circumradius of . In , following from the law of sines, we have .
Because and are tangents to the circumcircle , and . Thus, .
In , we have and . Thus, following from the law of cosines, we have
Following from the law of cosines,
Therefore,
Therefore, the answer is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution 1 by OmegaLearn.org
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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.