Difference between revisions of "2025 AIME II Problems/Problem 6"
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Circle <math>\omega_1</math> with radius <math>6</math> centered at point <math>A</math> is internally tangent at point <math>B</math> to circle <math>\omega_2</math> with radius <math>15</math>. Points <math>C</math> and <math>D</math> lie on <math>\omega_2</math> such that <math>\overline{BC}</math> is a diameter of <math>\omega_2</math> and <math>\overline{BC} \perp \overline{AD}</math>. The rectangle <math>EFGH</math> is inscribed in <math>\omega_1</math> such that <math>\overline{EF} \perp \overline{BC}</math>, <math>C</math> is closer to <math>\overline{GH}</math> than to <math>\overline{EF}</math>, and <math>D</math> is closer to <math>\overline{FG}</math> than to <math>\overline{EH}</math>, as shown. Triangles <math>\triangle DGF</math> and <math>\triangle CHG</math> have equal areas. The area of rectangle <math>EFGH</math> is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | Circle <math>\omega_1</math> with radius <math>6</math> centered at point <math>A</math> is internally tangent at point <math>B</math> to circle <math>\omega_2</math> with radius <math>15</math>. Points <math>C</math> and <math>D</math> lie on <math>\omega_2</math> such that <math>\overline{BC}</math> is a diameter of <math>\omega_2</math> and <math>\overline{BC} \perp \overline{AD}</math>. The rectangle <math>EFGH</math> is inscribed in <math>\omega_1</math> such that <math>\overline{EF} \perp \overline{BC}</math>, <math>C</math> is closer to <math>\overline{GH}</math> than to <math>\overline{EF}</math>, and <math>D</math> is closer to <math>\overline{FG}</math> than to <math>\overline{EH}</math>, as shown. Triangles <math>\triangle DGF</math> and <math>\triangle CHG</math> have equal areas. The area of rectangle <math>EFGH</math> is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
| − | + | <asy> | |
size(5cm); | size(5cm); | ||
defaultpen(fontsize(10pt)); | defaultpen(fontsize(10pt)); | ||
| Line 24: | Line 24: | ||
dot(H); | dot(H); | ||
| − | label(" | + | label("$A$", A, (.8, -.8)); |
| − | label(" | + | label("$B$", B, (.8, 0)); |
| − | label(" | + | label("$C$", C, (-.8, 0)); |
| − | label(" | + | label("$D$", D, (.4, .8)); |
| − | label(" | + | label("$E$", E, (.8, -.8)); |
| − | label(" | + | label("$F$", F, (.8, .8)); |
| − | label(" | + | label("$G$", G, (-.8, .8)); |
| − | label(" | + | label("$H$", H, (-.8, -.8)); |
| − | label(" | + | label("$\omega_1$", (9, -5)); |
| − | label(" | + | label("$\omega_2$", (-1, -13.5)); |
| − | + | </asy> | |
== Solution == | == Solution == | ||
Revision as of 00:59, 14 February 2025
Problem
Circle 
with radius 
centered at point 
is internally tangent at point 
to circle 
with radius 
. Points 
and 
lie on such that 
is a diameter of and 
. The rectangle 
is inscribed in such that 
, is closer to 
than to 
, and is closer to 
than to 
, as shown. Triangles 
and 
have equal areas. The area of rectangle is 
, where 
and 
are relatively prime positive integers. Find 
.
![[asy] size(5cm); defaultpen(fontsize(10pt)); pair A = (9, 0), B = (15, 0), C = (-15, 0), D = (9, 12), E = (9+12/sqrt(5), -6/sqrt(5)), F = (9+12/sqrt(5), 6/sqrt(5)), G = (9-12/sqrt(5), 6/sqrt(5)), H = (9-12/sqrt(5), -6/sqrt(5)); filldraw(G--H--C--cycle, lightgray); filldraw(D--G--F--cycle, lightgray); draw(B--C); draw(A--D); draw(E--F--G--H--cycle); draw(circle(origin, 15)); draw(circle(A, 6)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); label("$A$", A, (.8, -.8)); label("$B$", B, (.8, 0)); label("$C$", C, (-.8, 0)); label("$D$", D, (.4, .8)); label("$E$", E, (.8, -.8)); label("$F$", F, (.8, .8)); label("$G$", G, (-.8, .8)); label("$H$", H, (-.8, -.8)); label("$\omega_1$", (9, -5)); label("$\omega_2$", (-1, -13.5)); [/asy]](http://latex.artofproblemsolving.com/7/0/d/70d9d05c76e814d451d6889d6821d278137f2149.png)
