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6. Points (0, 0), (1, 1), (-1, -1), (2, 2), (-2, -2) are all on the same line which has equation y = x. The ants are always going from one point on the line to another point on the line. Also, as each ant goes from one point to the the next point, it walks a distance of sqrt(2). Since the ants started at the same point, the anthill at (0, 0), and walk in opposite directions along the same line, and each time they go from point to point, they go the same distance, they are always the same distance from the anthill (0, 0).

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