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Incircles

Source: 2024 AIME I #8

February 2, 2024
AMCAIMEAIME Igeometry

Problem Statement

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.
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