2021 AIME II Problems/Problem 14
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[hide]Problem
Let be an acute triangle with circumcenter and centroid . Let be the intersection of the line tangent to the circumcircle of at and the line perpendicular to at . Let be the intersection of lines and . Given that the measures of and are in the ratio the degree measure of can be written as where and are relatively prime positive integers. Find .
Solution 1
Let be the midpoint of . Because , and are cyclic, so is the center of the spiral similarity sending to , and . Because , it's easy to get from here.
~Lcz
Solution 2
Let be the midpoint of . Because we have cyclic and so ; likewise since we have cyclic and so . Now note that are collinear since is a median, so . But . Now letting we have and so .
See also
2021 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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