2026 AIME I Problem 12

Triangle \triangle ABC lies in plane \mathcal{P} with AB=6, AC=4, and \angle BAC=90^{\circ}. Let D be the reflection across \overline{BC} of the centroid of \triangle ABC. Four spheres, all on the same side of \mathcal{P}, have radii 1, 2, 3, and r and are tangent to \mathcal{P} at points A, B, C, and D, respectively. The four spheres are also each tangent to a second plane \mathcal{T} and are all on the same side of \mathcal{T}. The value of r can be written as \frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.