Difference between revisions of "2019 AMC 12A Problems/Problem 22"
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==Problem== | ==Problem== | ||
− | Circles <math>\omega</math> and <math>\gamma</math>, both centered at <math>O</math>, have radii <math>20</math> and <math>17</math>, respectively. Equilateral triangle <math>ABC</math>, whose interior lies in the interior of <math>\omega</math> but in the exterior of <math>\gamma</math>, has vertex <math>A</math> on <math>\omega</math>, and the line containing side <math>\overline{BC}</math> is tangent to <math>\gamma</math>. Segments <math>\overline{AO}</math> and <math>\overline{BC}</math> intersect at <math>P</math>, and <math>\dfrac{BP}{CP} = 3</math>. Then <math>AB</math> can be written in the form <math>\dfrac{m}{\sqrt{n}} - \dfrac{p}{\sqrt{q}}</math> for positive integers <math>m</math>, <math>n</math>, <math>p</math>, <math>q</math> with <math>\gcd(m,n) = \gcd(p,q) = 1</math>. What is <math>m+n+p+q</math>? | + | Circles <math>\omega</math> and <math>\gamma</math>, both centered at <math>O</math>, have radii <math>20</math> and <math>17</math>, respectively. Equilateral triangle <math>ABC</math>, whose interior lies in the interior of <math>\omega</math> but in the exterior of <math>\gamma</math>, has vertex <math>A</math> on <math>\omega</math>, and the line containing side <math>\overline{BC}</math> is tangent to <math>\gamma</math>. Segments <math>\overline{AO}</math> and <math>\overline{BC}</math> intersect at <math>P</math>, and <math>\dfrac{BP}{CP} = 3</math>. Then <math>AB</math> can be written in the form <math>\dfrac{m}{\sqrt{n}} - \dfrac{p}{\sqrt{q}}</math> for positive integers <math>m</math>, <math>n</math>, <math>p</math>, <math>q</math> with <math>\text{gcd}(m,n) = \text{gcd}(p,q) = 1</math>. What is <math>m+n+p+q</math>? |
<math>\phantom{}</math> | <math>\phantom{}</math> | ||
Revision as of 14:40, 14 October 2020
Contents
Problem
Circles and
, both centered at
, have radii
and
, respectively. Equilateral triangle
, whose interior lies in the interior of
but in the exterior of
, has vertex
on
, and the line containing side
is tangent to
. Segments
and
intersect at
, and
. Then
can be written in the form
for positive integers
,
,
,
with
. What is
?
Solution
Let be the point of tangency between
and
, and
be the midpoint of
. Note that
and
. This implies that
, and
. Thus,
.
If we let be the side length of
, then it follows that
and
. This implies that
, so
. Furthermore,
(because
) so this gives us the equation
to solve for the side length
, or
. Thus,
The problem asks for
.
Video Solution
https://www.youtube.com/watch?v=2eASfdhEyUE
See Also
2019 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.