2026 AIME II Problem 12

Consider a tetrahedron with two isosceles triangle faces with side lengths 5\sqrt{10}, \,5\sqrt{10}, and 10 and two isosceles triangle faces with side lengths 5\sqrt{10},\, 5\sqrt{10}, and 18. The four vertices of the tetrahedron lie on a sphere with center S, and the four faces of the tetrahedron are tangent to a sphere with center R. The distance RS can be written as \tfrac mn, 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.