2021 AIME II Problems/Problem 10

Problem

Two spheres with radii and one sphere with radius are each externally tangent to the other two spheres and to two different planes and . The intersection of planes and is the line . The distance from line to the point where the sphere with radius is tangent to plane is , where and are relatively prime positive integers. Find .

Solution

The centers of the three spheres form a 49-49-72 triangle. Consider the points at which the plane is tangent to the two bigger spheres; the line segment connecting these two points should be parallel to the 72 side of this triangle. Take its midpoint , which is 36 away from the midpoint of the 72 side , and connect these two midpoints.

Now consider the point at which the plane is tangent to the small sphere, and connect with the small sphere's tangent point . Extend through B until it hits the ray from through the center of the small sphere (convince yourself that these two intersect). Call this intersection , the center of the small sphere , we want to find .

By Pythagorus AC= , and we know . We know that must be parallel, using ratios we realize that . Apply Pythagorean theorem on triangle BCD; , so 312 + 23 =

-Ross Gao

-ARCTICTURN made slight edit to show the solution

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.