Arandomly chosen subject arrives for a study of exercise and fitness and begins walking on a treadmill at a brisk pace. which of the random variables described below is discrete

A discrete variable is a type of variable which contains a finite number of values. It usually takes its values from the set of whole numbers.

From the random variable described, the random variable that is descrete is "You measure the maximum heart rate in beats per minute".

This is because, heart rate cannot be partitioned it only takes counting number values. You either have 1 beat per minutes, 2 or 3 and so on beats per minute. But you cannot have 2.5 or 3.8 and so on beats per minute.
So heart rate cannot take decimal or fractional values and hence, it is a discrete random variable.