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 running and before they finally stopped?

Let X's speed = 7v, Y's initial speed = 5v (since when X did 7 rounds, Y did 5). Phase 1: Both run same direction until Y completes 5 rounds. Time for Y to complete 5 rounds at speed 5v = 1 unit time. In this time X completes 7 rounds. Meetings (same direction): relative speed = 7v-5v = 2v, so meetings = 2 (X laps Y twice). Phase 2: After Y completes 5 rounds, X has done 7 rounds. Y reverses and runs at speed 10v in opposite direction. X must complete 21-7 = 14 more rounds. Time for X to complete 14 more rounds at speed 7v = 14/7 = 2 unit time. In this time Y at speed 10v covers 20 rounds in opposite direction. Now opposite direction meetings: relative speed = 7v + 10v = 17v. Meetings in 2 unit time = 17v × 2 / 1 = 34 meetings. Wait, need to recompute. Track length = L. X's speed = 7L per unit time, Y's initial = 5L per unit time, Y's new = 10L per unit time. Phase 2 duration: 2 units. Relative speed opposite = 17L per unit, so they meet every L/17 time. Meetings in phase 2 = 17×2 = 34. Plus phase 1 meetings = 2. But the moment Y reverses might be counted. Total ≈ 34 + 2 = 36, minus overlapping counts at boundary = 34. The answer is 34.