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Difference between revisions of "2022 AIME II Problems/Problem 15"

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==Problem==
  
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Two externally tangent circles <math>\omega_1</math> and <math>\omega_2</math> have centers <math>O_1</math> and <math>O_2</math>, respectively. A third circle <math>\Omega</math> passing through <math>O_1</math> and <math>O_2</math> intersects <math>\omega_1</math> at <math>B</math> and <math>C</math> and <math>\omega_2</math> at <math>A</math> and <math>D</math>, as shown. Suppose that <math>AB = 2</math>, <math>O_1O_2 = 15</math>, <math>CD = 16</math>, and <math>ABO_1CDO_2</math> is a convex hexagon. Find the area of this hexagon.
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<asy>
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import geometry;
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size(10cm);
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point O1=(0,0),O2=(15,0),B=9*dir(30);
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circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B);
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point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0];
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filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black);
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draw(w1);
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draw(w2);
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draw(O1--O2,dashed);
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draw(o);
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dot(O1);
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dot(O2);
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dot(A);
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dot(D);
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dot(C);
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dot(B);
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label("$\omega_1$",8*dir(110),SW);
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label("$\omega_2$",5*dir(70)+(15,0),SE);
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label("$O_1$",O1,W);
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label("$O_2$",O2,E);
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label("$B$",B,N+1/2*E);
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label("$A$",A,N+1/2*W);
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label("$C$",C,S+1/4*W);
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label("$D$",D,S+1/4*E);
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label("$15$",midpoint(O1--O2),N);
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label("$16$",midpoint(C--D),N);
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label("$2$",midpoint(A--B),S);
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label("$\Omega$",o.C+(o.r-1)*dir(270));
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</asy>
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==Solution==
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==See Also==
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{{AIME box|year=2022|n=I|num-b=14|after=Last Problem}}
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{{MAA Notice}}

Revision as of 07:29, 18 February 2022

Problem

Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon. Find the area of this hexagon. [asy] import geometry; size(10cm); point O1=(0,0),O2=(15,0),B=9*dir(30); circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B); point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0]; filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black); draw(w1); draw(w2); draw(O1--O2,dashed); draw(o); dot(O1); dot(O2); dot(A); dot(D); dot(C); dot(B); label("$\omega_1$",8*dir(110),SW); label("$\omega_2$",5*dir(70)+(15,0),SE); label("$O_1$",O1,W); label("$O_2$",O2,E); label("$B$",B,N+1/2*E); label("$A$",A,N+1/2*W); label("$C$",C,S+1/4*W); label("$D$",D,S+1/4*E); label("$15$",midpoint(O1--O2),N); label("$16$",midpoint(C--D),N); label("$2$",midpoint(A--B),S); label("$\Omega$",o.C+(o.r-1)*dir(270)); [/asy]

Solution

See Also

2022 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Last Problem
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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