2021 AIME II Problems/Problem 14
Contents
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 .
Diagram
~MRENTHUSIASM (by Geometry Expressions)
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
In this solution, all angle measures are in degrees.
Let be the midpoint of so that and are collinear. Let and
Note that:
- Since quadrilateral is cyclic by the Converse of the Inscribed Angle Theorem.
It follows that as they share the same intercepted arc
- Since quadrilateral is cyclic by the supplementary opposite angles.
It follows that as they share the same intercepted arc
Together, we conclude that by AA, so
Next, we express in terms of By angle addition, we have Substituting back gives from which
For the sum of the interior angles of we get Finally, we obtain from which the answer is
~Constance-variance (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
Solution 3 (Guessing in the Last 3 Minutes, Unreliable)
Notice that looks isosceles, so we assume it's isosceles. Then, let and Taking the sum of the angles in the triangle gives so so the answer is
Solution 4 (Easy and Simple)
Video Solution 1
https://www.youtube.com/watch?v=zFH1Z7Ydq1s
Video Solution 2
https://www.youtube.com/watch?v=7Bxr2h4btWo
~Osman Nal
See also
2021 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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