Mock AIME 1 2010 Problems/Problem 13
Problem
Suppose is inscribed in circle . and are the feet of the altitude from to and to , respectively. Let be the intersection of lines and , let be the point of intersection of and line distinct from , and let be the foot of the perpendicular from to . Given that , , and , and that can be expressed in the form , where and are relatively prime positive integers and is an integer not divisible by the square of any prime, find the last three digits of .
Solution
Let . Because the problem gives us , we think to use the Law of Cosines in , which yields . Subtituting the values given by the problem, we get , which gives .
To find another expression for , we think of the formula . We know that the area of the triangle is . Substituting this in the previous equation for , we get that , so .
Setting these two expressions for equal to each other reveals that , so by the identity
is supplementary to , and is supplementary to , because is a cyclic quadrilateral. Thus, , so . Thus, , so our answer is .
See Also
Mock AIME 1 2010 (Problems, Source) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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