2022 AIME II Problems/Problem 15: Difference between revisions
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==Problem== | |||
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. | |||
<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== | |||
{{AIME box|year=2022|n=I|num-b=14|after=Last Problem}} | |||
{{MAA Notice}} | |||
Revision as of 07:29, 18 February 2022
Problem
Two externally tangent circles 
and 
have centers 
and 
, respectively. A third circle 
passing through and intersects at 
and 
and at 
and 
, as shown. Suppose that 
, 
, 
, and 
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]](http://latex.artofproblemsolving.com/7/4/a/74a9883958ea503b78cb2c07bc71aefe4d830ce9.png)
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 | ||
These problems are copyrighted © by the Mathematical Association of America. 
