X and Y are two runners who run for the same duration of time on the same circular track. They started running at the same time in the same direction with uniform speeds. When X completed 7 rounds, Y did exactly 5. After completing 5 rounds, Y changed his direction and started running in the opposite direction with speed which is double of his earlier speed. On the other hand, X continued to run with the same speed. They stopped running when X completed exactly 21 rounds. How many times did X and Y meet after they had started and before they finally stopped?<br/>

Let initial speeds be X:Y = 7:5. When Y completes 5 rounds, X completes 7 rounds; both are at the start point, so that is one meeting. Before that, since they run in the same direction, X gains on Y at rate 2, so he gains 1 full round once before this stage: one more meeting. After that point, Y reverses and doubles speed, so speeds become 7 and 10 in opposite directions; relative speed = 17. X has 14 rounds left to reach 21, which takes enough time for 34 relative laps. Starting together at the reversal point, they would meet 34 times including the final stopping instant; excluding that final instant gives 33 meetings in this phase. Total meetings = 1 + 1 + 33 = 35.