Difference between revisions of "2009 AIME I Problems/Problem 15"
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== Solution 3 == | == Solution 3 == | ||
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+ | First, we notice that triangle ABC is a scaled version of a 5-7-8 triangle (which has a 60 degree angle opposite the side with length 7). So <math>\angle{BAC} = 60^\circ{}</math>. Therefore, let <math>\angle{I_B AB} = \angle{I_B AD} = \alpha,</math> and <math>\angle{I_C AD} = \angle{I_C AC} = 30 - \alpha.</math> Therefore, in | ||
== See also == | == See also == | ||
{{AIME box|year=2009|n=I|num-b=14|after=Last Question}} | {{AIME box|year=2009|n=I|num-b=14|after=Last Question}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 09:53, 9 August 2019
Contents
[hide]Problem
In triangle , , , and . Let be a point in the interior of . Let and denote the incenters of triangles and , respectively. The circumcircles of triangles and meet at distinct points and . The maximum possible area of can be expressed in the form , where , , and are positive integers and is not divisible by the square of any prime. Find .
Solution 1
First, by Law of Cosines, we have so .
Let and be the circumcenters of triangles and , respectively. We first compute Because and are half of and , respectively, the above expression can be simplified to Similarly, . As a result
Therefore is constant (). Also, is or when is or . Let point be on the same side of as with ; is on the circle with as the center and as the radius, which is . The shortest distance from to is .
When the area of is the maximum, the distance from to has to be the greatest. In this case, it's . The maximum area of is and the requested answer is .
Solution 2
From Law of Cosines on , Now, Since and are cyclic quadrilaterals, it follows that Next, applying Law of Cosines on , By AM-GM, , so Finally, and the maximum area would be so the answer is .
Solution 3
First, we notice that triangle ABC is a scaled version of a 5-7-8 triangle (which has a 60 degree angle opposite the side with length 7). So . Therefore, let and Therefore, in
See also
2009 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Question | |
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.