9

Part of 2024 AIME

Problems(2)

Conic geo returns

Source: 2024 AIME I #9

A

B

C

D

be points in the coordinate plane on the hyperbola

\tfrac{x^{2}}{20}-\tfrac{y^{2}}{24}=1

ABCD

is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than

BD^{2}

for all such rhombi.

AMCAIMEAIME Iasymptotes

Another beautiful grid problem

Source: 2024 AIME II Problem 9

There is a collection of

25

indistinguishable black chips and

25

indistinguishable white chips. Find the number of ways to place some of these chips in

25

unit cells of a

5 \times 5

grid so that:
[*]each cell contains at most one chip, [*]all chips in the same row and all chips in the same column have the same color, [*]any additional chip placed on the grid would violate one or more of the previous two conditions.