As tension is applied to the rope, Weight A will be lifted first, until it is lifted to the ceiling. Then Weight B, and finally Weight A.
It helps to visualize Weight A as being massless. In that case, there would just be extra slack in the rope, and B&C would not move until the slack was taken in.
Ever lower milk prices were driving a dairy farmer to desperate measures. Two years ago, he tried "Beethoven for Bovines" in his barn and milk production dropped 2%. Last year he signed up for "hex the herd" where Genuine Santa Barbara WitchesTM remotely hexed your herd for health and higher production. (The ad had said its hexes were the cause of California's improved milk production, but it didn't seem to work in Wisconsin.) So this year he drove to town to consult the ultimate power source: a theoretical physicist. The physicist listened to his problem, asked a few questions, and then said he'd take the assignment, and that it would take only a few hours to solve the problem. A few weeks later, the physicist phoned the farmer, "I've got the answer. The solution turned out to be a bit more complicated than I thought and I'm presenting it at this afternoon's Theory Seminar". At the seminar the farmer finds a handful of people drinking tea and munching on cookies---none of whom looks like a farmer. As the talk begins the physicist approaches the blackboard and draws a big circle. "First, we assume a spherical cow..."
It helps to visualize Weight A as being massless. In that case, there would just be extra slack in the rope, and B&C would not move until the slack was taken in.