2011 IMO Shortlist C5

Let $m$ be a positive integer and consider a checkerboard consisting of $m$ by $m$ unit squares. At the midpoints of some of these unit squares there is an ant. At time 0 , each ant starts moving with speed 1 parallel to some edge of the checkerboard. When two ants moving in opposite directions meet, they both turn $90^{\circ}$ clockwise and continue moving with speed 1 . When more than two ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear. Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard or prove that such a moment does not necessarily exist.

令 $m$ 为正整数，并考虑由 $m$ × $m$ 个单位方块组成的棋盘。在其中一些单位方块的中点有一只蚂蚁。在时间 0 处，每只蚂蚁开始以速度 1 平行于棋盘的某些边缘移动。当两只向相反方向移动的蚂蚁相遇时，它们都会顺时针转动 $90^{\circ}$ 并继续以速度 1 移动。当超过两只蚂蚁相遇时，或者当两只沿垂直方向移动的蚂蚁相遇时，蚂蚁继续沿相遇前相同的方向移动。当一只蚂蚁到达棋盘边缘之一时，它会掉下来并且不会再次出现。考虑所有可能的起始位置，确定最后一只蚂蚁从棋盘上掉下来的最晚可能时刻，或者证明这样的时刻不一定存在。