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Problem

The velocity of a body increases with constant acceleration. Its motion is recorded between two points, $A$ and $B$, that are one metre apart, with $M$ being the midpoint of $AB$.

The body takes $p$ seconds to travel from $A$ to $M$ and $q$ seconds to travel from $M$ to $B$.

Given that the change in speed from $A$ to $B$, measured in metres per second, is integer, prove that $q$ cannot be integer.