Difference between revisions of "2022 AIME II Problems/Problem 15"
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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 08:29, 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.
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.