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− | ==Problem==
| + | #redirect [[2022 AMC 10B Problems/Problem 22]] |
− | Let <math>S</math> be the set of circles in the coordinate plane that are tangent to each of the three circles with equations <math>x^{2}+y^{2}=4</math>, <math>x^{2}+y^{2}=64</math>, and <math>(x-5)^{2}+y^{2}=3</math>. What is the sum of the areas of all circles in <math>S</math>?
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− | <math>\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad</math>
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− | ==Solution==
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− | <asy>
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− | import geometry;
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− | unitsize(0.5cm);
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− | void dc(pair x, pen p) {
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− | pair y = intersectionpoints(circle((0,0),8),(0,0)--1000*x)[0];
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− | draw(circle(x, abs(x-y)),p);
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− | }
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− | pair O1 = (0,0),O2=(5,0),P1=intersectionpoints(circle(O1,5),circle(O2,3+sqrt(3)))[0],P2=intersectionpoints(circle(O1,3),circle(O2,5+sqrt(3)))[0],P3=intersectionpoints(circle(O1,5),circle(O2,3-sqrt(3)))[0],P4=intersectionpoints(circle(O1,3),circle(O2,5-sqrt(3)))[0];
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− | draw(circle(O1,2));
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− | draw(circle(O1,8));
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− | draw(circle(O2,sqrt(3)));
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− | dc(P1,blue);
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− | dc(P2,red);
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− | dc(P3,darkgreen);
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− | dc(P4,brown);
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− | </asy>
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− | The circles match up as follows: Case 1 is brown, Case 2 is blue, Case 3 is green, and Case 4 is red.[/center]
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− | Let be circle , be circle , and be circle .
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− | All the circles in S are internally tangent to circle .
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− | There are four cases with two circle belonging to each:
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− | [*] and are internally tangent to S.
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− | [*] and are externally tangent to S.
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− | [*] is externally and Circle is internally tangent to S.
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− | [*] is internally and Circle is externally tangent to S.
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− | Consider Cases 1 and 4 together. Since circles and have the same center, the line connecting the center of and the center of will pass through both the tangency point of and and the tangency point of and . This line will be the diameter of and have length . Therefore the radius of in these cases is 5.
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− | Consider Cases 2 and 3 together. Similarly to Case 1 and 4, the line connecting the center of to the center of will pass through the tangency points. This time however, the diameter of will have length . Therefore, the radius of in these cases is 3.
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− | The set of circles S consists of 8 circles - 4 of which have radius 5 and 4 of which have radius 3.
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− | The total area of all circles in S is .
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− | -naman12
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− | {{AMC12 box|year=2022|ab=B|num-b=20|num-a=22}}
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− | {{AMC10 box|year=2022|ab=B|num-b=21|num-a=23}}
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− | {{MAA Notice}}
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