2020 AIME II Problems/Problem 7
Problem
Two congruent right circular cones each with base radius 
and height 
have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance from the base of each cone. A sphere with radius 
lies withing both cones. The maximum possible value of 
is 
, where 
n and 
are relatively prime positive integers. Find 
.
Solution
Take the cross-section of the plane of symmetry formed by the two cones. Let the point where the bases intersect be the origin, 
, and the bases form the positive 
and 
axes. Then label the vertices of the region enclosed by the two triangles as 
in a clockwise manner. We want to find the radius of the inscribed circle of 
. By symmetry, the center of this circle must be 
. 
can be represented as 
Using the point-line distance formula, ![\[r^2=\frac{15^2}{(\sqrt{8^2+3^2})^2}=\frac{225}{73}\]](http://latex.artofproblemsolving.com/6/b/7/6b7cba3ed99e0adaec3632080a57f9f390e95723.png)
This implies our answer is 
. ~mn28407
See Also
| 2020 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 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. 
