(Also: this is not a problem concerned with the physics of rubber or biology of ants. This ant is immortal, the rubber infinitely stretchy, etc.)
Which is great, except that when the original concept is framed to the reader, the simple word problem doesn't specifically express that both ends are in motion.
If both ends are in motion, then the ant's velocity changes. The ant might move 1 centimeter per second, relative to a fixed point in space, but it isn't moving 1 centimeter per second, relative to it's origin, POINT A.
The motion of the rubber band is influencing the speed of the ant, relative to the band itself. Since the ant moves toward POINT B, while POINT A simultaneously accelerates away from the ant, the ant is moving faster than 1 centimeter per second, with each second spent traveling away from POINT A.
Once this detail is added, it completely changes the premise of the problem. But the manner in which the problem is originally expressed is vague and incomplete, misleading the reader (perhaps deliberately so).
Why would anyone claim the ant is moving 1 centimeter per second, RELATIVE TO THE ONLY OTHER FRAME OF REFERENCE IN THE PROBLEM (the rubber band), when its motion is clearly not 1 centimeter per second?
In that case, the ant is only moving at a fixed speed, toward the other end (100,000 times slower than the end's expansion), which will forever be too far away.