Difference between revisions of "2024 AIME II Problems/Problem 12"
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− | Let <math>C = (\tfrac18,\tfrac{3\sqrt3}8)</math>. Draw a line through <math>C</math> intersecting the <math>x</math>-axis at <math>A'</math> and the <math>y</math>-axis at <math>B'</math>. We shall show that <math>A'B' \ge 1</math>, and that equality only holds when <math>A'=A</math> and <math>B'=B</math>. | + | Let <math>C = (\tfrac18,\tfrac{3\sqrt3}8)</math>. <s>this is sus, furaken randomly guessed C and proceeded to prove it works</s> Draw a line through <math>C</math> intersecting the <math>x</math>-axis at <math>A'</math> and the <math>y</math>-axis at <math>B'</math>. We shall show that <math>A'B' \ge 1</math>, and that equality only holds when <math>A'=A</math> and <math>B'=B</math>. |
Let <math>\theta = \angle OA'C</math>. Draw <math>CD</math> perpendicular to the <math>x</math>-axis and <math>CE</math> perpendicular to the <math>y</math>-axis as shown in the diagram. Then | Let <math>\theta = \angle OA'C</math>. Draw <math>CD</math> perpendicular to the <math>x</math>-axis and <math>CE</math> perpendicular to the <math>y</math>-axis as shown in the diagram. Then |
Revision as of 06:09, 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
.