2021 CIME I Problems/Problem 14
Let be an acute triangle with orthocenter and circumcenter . The tangent to the circumcircle of at intersects lines and at and , and . Let line intersect at . Suppose that , and for positive integers where is not divisible by the square of any prime. Find .
Solution by TheUltimate123
Let be the orthocenter of , and let , be the feet of the altitudes from . Also let be the antipode of on the circumcircle and let , as shown below: Disregarding the condition , we contend:
In general, is cyclic.
Recall that , so the claim follows from Reims' theorem on
With , it follows that is an isosceles trapezoid. In particular, and . Since , we have But note that , so i.e.\ . We are given , and by the law of sines, , so , and , so .
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