BY-MO 2019 Problem 98

Andrey and Sasha play the game, making moves alternate. On his turn, Andrey marks on the plane an arbitrary point that has not yet been marked. After that, Sasha colors this point in one of two colors: white and black. Sasha wins if after his move it is impossible to draw a line such that all white points lie in one half-plane, while all black points lie in another half-plane with respect to this line.a) Prove that Andrey can make moves in such a way that Sasha will never win.b) Suppose that Andrey can mark only integer points on the Cartesian plane. Can Sasha guarantee himself a win regardless of Andrey's moves?

(N. Naradzetski)