Wagon collision

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This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov.

Puzzle

Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?

Help

Solution
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References

Ordinary Differential Equations - V. I. Arnold's book on ODEs.