“Lewis and Carol travel together on a road from A to B, then return on the same road, with the entire trip taking 3 hours. Sometimes that road goes uphill, sometimes downhill, and sometimes it is level. When the road goes uphill, their rate is 40 mph; downhill their rate is 60 mph; on level road, their rate is *x* mph.”

Even if you were given a numerical value for *x*, the distance from A to B would (in most cases) not be uniquely determined. But there is one value for *x* that would determine that distance uniquely. Compute this value of *x* [Note: uphill going is downhill returning!]

Credits – Southern California ARML, Oct. 2009, Team round

Original post here.

Solution follows.

[spoiler]

Let be the distance spent on level road, and let be the distance spent on sloped road (which way, uphill or downhill, doesn’t matter because it’ll be reversed when they turn back).

This equation can be simplified to:

Or, multiplying both sides by 24:

The only value of for which can be uniquely defined is .

[/spoiler]

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