Difference between revisions of "2023 AMC 12A Problems/Problem 18"

2023 AMC 10A (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 10 Problems and Solutions

2023 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
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

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 {{duplicate|[[2023 AMC 10A Problems/Problem 22|2023 AMC 10A #22]] and [[2023 AMC 12A Problems/Problem 18|2023 AMC 12A #18]]}}
-==Problem==
-Circle <math>C_1</math> and <math>C_2</math> each have radius <math>1</math>, and the distance between their centers is <math>\frac{1}{2}</math>. Circle <math>C_3</math> is the largest circle internally tangent to both <math>C_1</math> and <math>C_2</math>. Circle <math>C_4</math> is internally tangent to both <math>C_1</math> and <math>C_2</math> and externally tangent to <math>C_3</math>. What is the radius of <math>C_4</math>?
-<asy>
-import olympiad;
-size(10cm);
-draw(circle((0,0),0.75));
-draw(circle((-0.25,0),1));
-draw(circle((0.25,0),1));
-draw(circle((0,6/7),3/28));
-pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118);
-dot(B^^C);
-draw(B--E, dashed);
-draw(C--F, dashed);
-draw(B--C);
-label("$C_4$", D);
-label("$C_1$", (-1.375, 0));
-label("$C_2$", (1.375,0));
-label("$\frac{1}{2}$", (0, -.125));
-label("$C_3$", (-0.4, -0.4));
-label("$1$", (-.85, 0.70));
-label("$1$", (.85, -.7));
-import olympiad;
-markscalefactor=0.005;
-</asy>
-<math>\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}</math>
 ==Video Solution by Little Fermat==

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