Problem:
Let △ABC be an acute triangle with circumcenter O and centroid G. Let X be the intersection of the line tangent to the circumcircle of △ABC at A and the line perpendicular to GO at G. Let Y be the intersection of lines XG and BC. Given that the measures of ∠ABC,∠BCA, and ∠XOY are in the ratio 13:2:17, the degree measure of ∠BAC can be written as nm, where m and n are relatively prime positive integers. Find m+n.
Solution:
Answer (592):
There is a real number x>0 such that 13x=∠ABC>∠BCA=2x. Let M be the midpoint of BC, and let T be the intersection of lines XA and BC.
Because ∠OAX=∠OMY=∠OGY=90∘, both OGYM and OGAX are cyclic. Then it follows that