Difference between revisions of "2017 AIME II Problems/Problem 12"
The turtle (talk | contribs) (Created page with "<math>\textbf{Problem 12}</math> Circle <math>C_0</math> has radius <math>1</math>, and the point <math>A_0</math> is a point on the circle. Circle <math>C_1</math> has radius...") |
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Circle <math>C_0</math> has radius <math>1</math>, and the point <math>A_0</math> is a point on the circle. Circle <math>C_1</math> has radius <math>r<1</math> and is internally tangent to <math>C_0</math> at point <math>A_0</math>. Point <math>A_1</math> lies on circle <math>C_1</math> so that <math>A_1</math> is located <math>90^{\circ}</math> counterclockwise from <math>A_0</math> on <math>C_1</math>. Circle <math>C_2</math> has radius <math>r^2</math> and is internally tangent to <math>C_1</math> at point <math>A_1</math>. In this way a sequence of circles <math>C_1,C_2,C_3,\cdots</math> and a sequence of points on the circles <math>A_1,A_2,A_3,\cdots</math> are constructed, where circle <math>C_n</math> has radius <math>r^n</math> and is internally tangent to circle <math>C_{n-1}</math> at point <math>A_{n-1}</math>, and point <math>A_n</math> lies on <math>C_n</math> <math>90^{\circ}</math> counterclockwise from point <math>A_{n-1}</math>, as shown in the figure below. There is one point <math>B</math> inside all of these circles. When <math>r = \frac{11}{60}</math>, the distance from the center <math>C_0</math> to <math>B</math> is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | Circle <math>C_0</math> has radius <math>1</math>, and the point <math>A_0</math> is a point on the circle. Circle <math>C_1</math> has radius <math>r<1</math> and is internally tangent to <math>C_0</math> at point <math>A_0</math>. Point <math>A_1</math> lies on circle <math>C_1</math> so that <math>A_1</math> is located <math>90^{\circ}</math> counterclockwise from <math>A_0</math> on <math>C_1</math>. Circle <math>C_2</math> has radius <math>r^2</math> and is internally tangent to <math>C_1</math> at point <math>A_1</math>. In this way a sequence of circles <math>C_1,C_2,C_3,\cdots</math> and a sequence of points on the circles <math>A_1,A_2,A_3,\cdots</math> are constructed, where circle <math>C_n</math> has radius <math>r^n</math> and is internally tangent to circle <math>C_{n-1}</math> at point <math>A_{n-1}</math>, and point <math>A_n</math> lies on <math>C_n</math> <math>90^{\circ}</math> counterclockwise from point <math>A_{n-1}</math>, as shown in the figure below. There is one point <math>B</math> inside all of these circles. When <math>r = \frac{11}{60}</math>, the distance from the center <math>C_0</math> to <math>B</math> is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
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− | + | ==Solution== | |
<math>\boxed{110}</math> | <math>\boxed{110}</math> | ||
+ | |||
+ | =See Also= | ||
+ | {{AIME box|year=2017|n=II|num-b=11|num-a=13}} | ||
+ | {{MAA Notice}} |
Revision as of 12:01, 23 March 2017
Problem
Circle has radius
, and the point
is a point on the circle. Circle
has radius
and is internally tangent to
at point
. Point
lies on circle
so that
is located
counterclockwise from
on
. Circle
has radius
and is internally tangent to
at point
. In this way a sequence of circles
and a sequence of points on the circles
are constructed, where circle
has radius
and is internally tangent to circle
at point
, and point
lies on
counterclockwise from point
, as shown in the figure below. There is one point
inside all of these circles. When
, the distance from the center
to
is
, where
and
are relatively prime positive integers. Find
.
[asy]
draw(Circle((0,0),125));
draw(Circle((25,0),100));
draw(Circle((25,20),80));
draw(Circle((9,20),64));
dot((125,0));
label("
",(125,0),E);
dot((25,100));
label("
",(25,100),SE);
dot((-55,20));
label("
",(-55,20),E);
[/asy]
Solution
See Also
2017 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.