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# Three cars

## Question

There are three cars driving on a track.  The track is a perfect circle (circumference unknown) and is not wide enough to allow any car to pass another.  Currently, the lead car is going 55 MPH and the last car is going 45 MPH.  While the car in the middle is going some speed between.  At this moment there is x miles between the lead car and the middle car and x miles between the middle car and the slowest car where x is not 0 or 1 miles.  If the car's maintained their speed up to the point where the lead car caught the slowest car (then everyone stops), would there ever be a point and time where the distance between any two pairs is again x miles (the pairs must be x distance apart at the same time)?

## 5 answers to this question

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if the car in the middle is going at 50 MPH, so the distance between them will increase by (x) miles each hour, for a fixed time, then it will decrease between the lead car and the slowest car untill they meet.

so there would be no point where the distance between any two cars is (x) miles again.

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Is this solvable without knowing circumfrence?

Edited by JIntorcio

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Agree with JIntorcio.

The restriction that x is not 1 mile is innocuous, since that value has no inferable relation to the length of the track.

A standard approach with an unspecified parameter is to assign it a value, say 10 in this case, inferring that the answer does not depend on the value. (else it would be given.) Best example of this is the hole-in-the-sphere problem.

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Unless the middle car's speed needs to be in a closed interval (possibly including 45 and 55) rather than an open one (where it's strictly greater than 45 and less than 55). If the distance between, say, the middle car and the initially last car remains consistently x, then the first car will eventually be x distance away from catching the last car and there will be two pairs x apart.

Or if other weird things are going on like the track is not circular and distance is measured in Euclidian space rather than distance along the track.

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Fair point

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