2010 AIME I Problems/Problem 15
Problem
In with , , and , let be a point on such that the incircles of and have equal radii. Let and be positive relatively prime integers such that . Find .
Solution
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Let and so . Let the incenters of and be and respectively, and their equal inradii be . From , we find that
Let the incircle of meet at and the incircle of meet at . Then note that is a rectangle. Also, is right because and are the angle bisectors of and respectively and . By properties of tangents to circles and . Now notice that the altitude of to is of length , so by similar triangles we find that (3). Equating (3) with (1) and (2) separately yields
and adding these we have
See also
2010 AIME I (Problems • Answer Key • Resources) | ||
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