2020 AIME II Problems/Problem 7
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 .
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
|2020 AIME II (Problems • Answer Key • Resources)|
|1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15|
|All AIME Problems and Solutions|