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AIME 2024 / AIME 2024 train ec28d7b6-51c8-5477-a0a2-89eba61dc380

Problem

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Problem

Eight circles of radius 3434 are sequentially tangent, and two of the circles are tangent to ABAB and BCBC of triangle ABCABC, respectively. 20242024 circles of radius 11 can be arranged in the same manner. The inradius of triangle ABCABC can be expressed as mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
Plain-text mathematical notation (without MathML)
Eight circles of radius 34 are sequentially tangent, and two of the circles are tangent to AB and BC of triangle ABC, respectively. 2024 circles of radius 1 can be arranged in the same manner. The inradius of triangle ABC can be expressed as (m)/(n), where m and n are relatively prime positive integers. Find m+n.
Original LaTeX notation
Eight circles of radius $34$ are sequentially tangent, and two of the circles are tangent to $AB$ and $BC$ of triangle $ABC$, respectively. $2024$ circles of radius $1$ can be arranged in the same manner. The inradius of triangle $ABC$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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Source and history

task_id
ec28d7b6-51c8-5477-a0a2-89eba61dc380
task_key
default--train--ec28d7b6-51c8-5477-a0a2-89eba61dc380
task_revision_id
3
config / split
default / train

Official source

initial import