2024 AIME I Problems/Problem 10
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[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,(see Symmedian and tangents )
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)
Solution 6
Note that since P is a symmedian, are harmonic. As a result, . As a result, . Call . Then, . Since , . Use LOC to find . Finish with Ptolemy on ABPC, and finish to get .
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 |
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