10

Part of 2022 AIME Problems

Problems(2)

detoasty3-esque geo

Source: 2022 AIME I #10

Three spheres with radii

11

13

19

are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at

A

B

C

, respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that

AB^2 = 560

AC^2

AMCAIME3 big balls

choosechoosechoosechoosechoosechoose

Source: 2022 AIME II #10

Find the remainder when

\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}

1000