USAMO 2004 — Days 1 and 2 Question 4

Alice and Bob play a game on a $6$ by $6$ grid. On each turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square. Alice goes first and the players alternate. When the grid is full, the square with the greatest number in each row is colored black. Alice wins if a line can be drawn from the top of the grid to the bottom that stays in black squares; Bob wins otherwise. If two squares share a vertex, the line may pass from one to the other while staying in those two squares. Find, with proof, a winning strategy for one player.

Proposed by Melanie Wood (credited in the supplied source).