2020 AIME II Problems/Problem 7
Contents
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,
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.
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.