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Difference between revisions of "2024 AIME II Problems/Problem 12"

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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
 
~Furaken
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 +
==Solution 2==
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 +
<math>y=-\tan \theta x+\sin \theta=-\sqrt{3}x+\frac{\sqrt{3}}{2}, x=\frac{\sqrt{3}-2\sin \theta}{2\sqrt{3}-2\tan \theta}</math>
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 +
When <math>\theta=\frac{\pi}{3}</math>, the limit of <math>x=\frac{1}{8}\implies y=\frac{3\sqrt{3}}{8}\implies \frac{7}{16}=OC^2, \boxed{023}</math>
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~Bluesoul

Revision as of 11:02, 9 February 2024

Let $O(0,0),A(\tfrac{1}{2},0),$ and $B(0,\tfrac{\sqrt{3}}{2})$ be points in the coordinate plane. Let $\mathcal{F}$ be the family of segments $\overline{PQ}$ of unit length lying in the first quadrant with $P$ on the $x$-axis and $Q$ on the $y$-axis. There is a unique point $C$ on $\overline{AB},$ distinct from $A$ and $B,$ that does not belong to any segment from $\mathcal{F}$ other than $\overline{AB}$. Then $OC^2=\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution 1

By Furaken [asy] pair O=(0,0); pair X=(1,0); pair Y=(0,1); pair A=(0.5,0); pair B=(0,sin(pi/3)); dot(O); dot(X); dot(Y); dot(A); dot(B); draw(X--O--Y); draw(A--B); label("$B'$", B, W); label("$A'$", A, S); label("$O$", O, SW); pair C=(1/8,3*sqrt(3)/8); dot(C); pair D=(1/8,0); dot(D); pair E=(0,3*sqrt(3)/8); dot(E); label("$C$", C, NE); label("$D$", D, S); label("$E$", E, W); draw(D--C--E); [/asy]

Let $C = (\tfrac18,\tfrac{3\sqrt3}8)$. this is sus, furaken randomly guessed C and proceeded to prove it works Draw a line through $C$ intersecting the $x$-axis at $A'$ and the $y$-axis at $B'$. We shall show that $A'B' \ge 1$, and that equality only holds when $A'=A$ and $B'=B$.

Let $\theta = \angle OA'C$. Draw $CD$ perpendicular to the $x$-axis and $CE$ perpendicular to the $y$-axis as shown in the diagram. Then \[8A'B' = 8CA' + 8CB' = \frac{3\sqrt3}{\sin\theta} + \frac{1}{\cos\theta}\] By some inequality (i forgor its name), \[\left(\frac{3\sqrt3}{\sin\theta} + \frac{1}{\cos\theta}\right) \cdot \left(\frac{3\sqrt3}{\sin\theta} + \frac{1}{\cos\theta}\right) \cdot (\sin^2\theta + \cos^2\theta) \ge (3+1)^3 = 64\] We know that $\sin^2\theta + \cos^2\theta = 1$. Thus $\tfrac{3\sqrt3}{\sin\theta} + \tfrac{1}{\cos\theta} \ge 8$. Equality holds if and only if \[\frac{3\sqrt3}{\sin\theta} : \frac{1}{\cos\theta} = \frac{3\sqrt3}{\sin\theta} : \frac{1}{\cos\theta} = \sin^2\theta : \cos^2\theta\] which occurs when $\theta=\tfrac\pi3$. Guess what, $\angle OAB$ happens to be $\tfrac\pi3$, thus $A'=A$ and $B'=B$. Thus, $AB$ is the only segment in $\mathcal{F}$ that passes through $C$. Finally, we calculate $OC^2 = \tfrac1{64} + \tfrac{27}{64} = \tfrac7{16}$, and the answer is $\boxed{023}$. ~Furaken

Solution 2

$y=-\tan \theta x+\sin \theta=-\sqrt{3}x+\frac{\sqrt{3}}{2}, x=\frac{\sqrt{3}-2\sin \theta}{2\sqrt{3}-2\tan \theta}$

When $\theta=\frac{\pi}{3}$, the limit of $x=\frac{1}{8}\implies y=\frac{3\sqrt{3}}{8}\implies \frac{7}{16}=OC^2, \boxed{023}$

~Bluesoul

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