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 06:13, 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