MathDB
2021 Team P14

Source:

March 2, 2021
combinatorics

Problem Statement

Let SS be the set of lattice points (x,y)∈Z2(x,y) \in \mathbb{Z}^2 such that −10≤x,y≤10-10\leq x,y \leq 10. Let the point (0,0)(0,0) be OO. Let Scotty the Dog's position be point PP, where initially P=(0,1)P=(0,1). At every second, consider all pairs of points C,D∈SC,D \in S such that neither CC nor DD lies on line OPOP, and the area of quadrilateral OCPDOCPD (with the points going clockwise in that order) is 11. Scotty finds the pair C,DC,D maximizing the sum of the yy coordinates of CC and DD, and randomly jumps to one of them, setting that as the new point PP. After 5050 such moves, Scotty ends up at point (1,1)(1, 1). Find the probability that he never returned to the point (0,1)(0,1) during these 5050 moves.
Proposed by David Tang
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