MathDB

8

Part of 2024 AIME

Problems(2)

Incircles

Source: 2024 AIME I #8

2/2/2024
Eight circles of radius 3434 can be placed tangent to side BC‾\overline{BC} of △ABC\triangle ABC such that the first circle is tangent to AB‾\overline{AB}, subsequent circles are externally tangent to each other, and the last is tangent to AC‾\overline{AC}. Similarly, 20242024 circles of radius 11 can also be placed along BC‾\overline{BC} in this manner. The inradius of △ABC\triangle ABC is mn\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
AMCAIMEAIME Igeometry
Torus Geo Showed Up on AIME

Source: 2024 AIME II #8

2/8/2024
Torus T\mathcal T is the surface produced by revolving a circle with radius 33 around an axis in the plane a distance 66 from the center of the circle. When a sphere of radius 1111 rests inside T\mathcal T, it is internally tangent to T\mathcal T along a circle with radius rir_{i}, and when it rests outside T\mathcal T, it is externally tangent along a circle with radius ror_{o}. The difference ri−ro=mnr_{i}-r_{o}=\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
AMCAIMEAIME II