2021 AIME II Problems/Problem 14

Problem

Let $\Delta ABC$ be an acute triangle with circumcenter $O$ and centroid $G$ . Let $X$ be the intersection of the line tangent to the circumcircle of $\Delta ABC$ 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 $\angle ABC, \angle BCA,$ and $\angle XOY$ are in the ratio $13 : 2 : 17,$ the degree measure of $\angle BAC$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution 1

Let $M$ be the midpoint of $BC$ . Because $\angle{OAX}=\angle{OGX}=\angle{OGY}=\angle{OMY}=90^o$ , $AXOG$ and $OMYG$ are cyclic, so $O$ is the center of the spiral similarity sending $AM$ to $XY$ , and $\angle{XOY}=\angle{AOM}$ . Because $\angle{AOM}=2\angle{BCA}+\angle{BAC}$ , it's easy to get $\frac{585}{7} \implies \boxed{592}$ from here.

~Lcz

Solution 2

Let $M$ be the midpoint of $BC$ . Because $\angle OAX = \angle OGX = 90$ we have $AXOG$ cyclic and so $\angle GXO = \angle OAG$ ; likewise since $\angle OMY = \angle OGY = 90$ we have $OMYG$ cyclic and so $\angle OYG = \angle OMG$ . Now note that $A, G, M$ are collinear since $\overline{AM}$ is a median, so $\triangle AOM \sim \triangle XOY$ . But $\angle AOM = \angle AOB + \angle BOM = 2 \angle C + \angle A$ . Now letting $\angle C = 2k, \angle B = 13k, \angle AOM = \angle XOY = 17k$ we have $\angle A = 13k$ and so $\angle A = \frac{585}{7} \implies \boxed{592}$ .

2021 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2021 AIME II Problems/Problem 14

Contents

Problem

Solution 1

Solution 2

See also