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2026
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?
Explanation
When X completes 21 rounds, Y, originally at 5 rounds, switches direction and runs at double speed. They meet every 1 round of X, or every 7 rounds of Y (due to their speed ratio of 7:5), leading to 21 meetings while X completes 21 rounds, resulting in a total of while in the same direction and while in opposite directions, totaling . However, since they met during the original 7 rounds as well, the total indeed is .
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