Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2025 AIME II Problems/Problem 6"

(→‎Problem)
(→‎Problem)
Line 2: Line 2:
 
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);         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((0,0), 15));         draw(circle(A, 6));         dot(A);         dot(B);         dot(C);         dot(D);         dot(E);         dot(F);         dot(G);         dot(H);         label("<math>A</math>", A, (.8, -.8));         label("<math>B</math>", B, (.8, 0));         label("<math>C</math>", C, (-.8, 0));         label("<math>D</math>", D, (.4, .8));         label("<math>E</math>", E, (.8, -.8));         label("<math>F</math>", F, (.8, .8));         label("<math>G</math>", G, (-.8, .8));         label("<math>H</math>", H, (-.8, -.8));         label("<math>\omega_1</math>", (9, -5));         label("<math>\omega_2</math>", (-1, -13.5)); [/asy]
+
[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("<math>A</math>", A, (.8, -.8));
 +
label("<math>B</math>", B, (.8, 0));
 +
label("<math>C</math>", C, (-.8, 0));
 +
label("<math>D</math>", D, (.4, .8));
 +
label("<math>E</math>", E, (.8, -.8));
 +
label("<math>F</math>", F, (.8, .8));
 +
label("<math>G</math>", G, (-.8, .8));
 +
label("<math>H</math>", H, (-.8, -.8));
 +
label("<math>\omega_1</math>", (9, -5));
 +
label("<math>\omega_2</math>", (-1, -13.5));
 +
[/asy]
  
 
== Solution ==
 
== Solution ==

Revision as of 00:57, 14 February 2025

Problem

Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and $\overline{BC} \perp \overline{AD}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such that $\overline{EF} \perp \overline{BC}$, $C$ is closer to $\overline{GH}$ than to $\overline{EF}$, and $D$ is closer to $\overline{FG}$ than to $\overline{EH}$, as shown. Triangles $\triangle DGF$ and $\triangle CHG$ have equal areas. The area of rectangle $EFGH$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

[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]

Solution

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2025_AIME_II_Problems/Problem_6&oldid=242453"