Difference between revisions of "2024 AIME II Problems/Problem 12"
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<cmath>\frac{3\sqrt3}{\sin\theta} : \frac{1}{\cos\theta} = \frac{3\sqrt3}{\sin\theta} : \frac{1}{\cos\theta} = \sin^2\theta : \cos^2\theta</cmath> | <cmath>\frac{3\sqrt3}{\sin\theta} : \frac{1}{\cos\theta} = \frac{3\sqrt3}{\sin\theta} : \frac{1}{\cos\theta} = \sin^2\theta : \cos^2\theta</cmath> | ||
which occurs when <math>\theta=\tfrac\pi3</math>. Guess what, <math>\angle OAB</math> ''happens'' to be <math>\tfrac\pi3</math>, thus <math>A'=A</math> and <math>B'=B</math>. Thus, <math>AB</math> is the only segment in <math>\mathcal{F}</math> that passes through <math>C</math>. Finally, we calculate <math>OC^2 = \tfrac1{64} + \tfrac{27}{64} = \tfrac7{16}</math>, and the answer is <math>\boxed{023}</math>. | which occurs when <math>\theta=\tfrac\pi3</math>. Guess what, <math>\angle OAB</math> ''happens'' to be <math>\tfrac\pi3</math>, thus <math>A'=A</math> and <math>B'=B</math>. Thus, <math>AB</math> is the only segment in <math>\mathcal{F}</math> that passes through <math>C</math>. Finally, we calculate <math>OC^2 = \tfrac1{64} + \tfrac{27}{64} = \tfrac7{16}</math>, and the answer is <math>\boxed{023}</math>. | ||
+ | ~Furaken |
Revision as of 05:13, 9 February 2024
Let and be points in the coordinate plane. Let be the family of segments of unit length lying in the first quadrant with on the -axis and on the -axis. There is a unique point on distinct from and that does not belong to any segment from other than . Then , where and are relatively prime positive integers. Find .
Solution 1
By Furaken
Let . this is sus, furaken randomly guessed C and proceeded to prove it works Draw a line through intersecting the -axis at and the -axis at . We shall show that , and that equality only holds when and .
Let . Draw perpendicular to the -axis and perpendicular to the -axis as shown in the diagram. Then By some inequality (i forgor its name), We know that . Thus . Equality holds if and only if which occurs when . Guess what, happens to be , thus and . Thus, is the only segment in that passes through . Finally, we calculate , and the answer is . ~Furaken