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I am reminded of this question that appeared in a regional Physics Olympiad I had appeared. I was wondering if there is a "mathematical" way of doing it.

If you start from a point $A$ at midnight along a specified path down a mountain and reach a specified point $B$ exactly 24 hours later. At this point you reverse your direction of travel and travel along the same path to point $A$ and reach $A$ exactly 24 hours later. At any point of time during these two days, your velocity can be positive, negative or zero (and of course less than $c$). Prove that there exists at least one point along the path where you were at the same time on day 1 and day 2.

It "seems" like an application of intermediate value theorem for some appropriately defined function, but I am not sure (though I will tag it as calculus for the moment). Any ideas?

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This a classic!

A "layman" proof is to consider you and your ghost from the previous day.

When you turn back, let your ghost of the previous day start from point $A$. At some point, you and ghost must meet!

In stricter mathematical terms:

You have two continuous functions, which measure the distance from point $A$ with time (in days say). Let the distance from $A$ to $B$ be 1km.

You have that $f(0) = 0, f(1) = 1, g(0) = 1, g(1) = 0$. $f$ is the function for day 1 and $g$ is the function for day 2.

Now $h(x) = f(x) - g(x)$ is continuous with $h(0) < 0$ and $h(1) > 0$ and so must be $0$ at some point in between, using the intermediate value theorem.

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    Nice one. A problem based on the same idea was asked to me in an interview. You have a thread and if you light the thread at one end the thread will burn in 1 hour. The thread has a non-uniform rate of burning. How do you use this to measure 30 minutes? The answer is you light up both ends and the argument is similar to this problem.2010-11-26
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    Thanks. That was what I was trying, except I was using the actual distance traveled as my function and wasn't getting anywhere.2010-11-26