RU-ALLRUSSIAN 2009 Problem 114

Given a set $M$ of points $(x,y)$ with integral coordinates satisfying $x^2 + y^2\leq 10^{10}$ . Two players play a game. One of them marks a point on his first move. After this, on each move the moving player marks a point, which is not yet marked and joins it with the previous marked point. Players are not allowed to mark a point symmetrical to the one just chosen. So, they draw a broken line. The requirement is that lengths of edges of this broken line must strictly increase. The player, which can not make a move, loses. Who have a winning strategy?