Interesting proof about a running track

[Thales12345]

GIVEN: Three runners start at the start line of the running track and run at a constant, but different pace, in the same direction.

TO PROVE: If they run long enough, there will come a point where 2 runners are at least a third of the way from the third runner's lane.

Does anyone have a suggestion on how to get started on this? This one is so hard!

 

[lev888]

GIVEN: Three runners start at the start line of the running track and run at a constant, but different pace, in the same direction.

TO PROVE: If they run long enough, there will come a point where 2 runners are at least a third of the way from the third runner's lane.

Does anyone have a suggestion on how to get started on this? This one is so hard!

This is a problem, not a proof - we don't know whether the proof will be interesting.
The problem statement doesn't seem very precise to me. How can 2 runners be a certain distance from the 3rd runner's _lane_? Do you mean "from the 3rd runner"? And what is the "way", which is the basis for "a third"?

 

[Otis]

Hi Thales. Where did you find this exercise? Have you typed it exactly as shown? Did you see any diagram? (Running tracks vary.)

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[Thales12345]

I found this proof (or according to Lev 888 “problem”) in an old math book by a friend of my father's. However, there was no solution in the book, but I am curious about how this should be solved ...

And yes, the question was also written in the book like that, so without numbers or diagrams...

 

[Dr.Peterson]

GIVEN: Three runners start at the start line of the running track and run at a constant, but different pace, in the same direction.

TO PROVE: If they run long enough, there will come a point where 2 runners are at least a third of the way from the third runner's lane.

Does anyone have a suggestion on how to get started on this? This one is so hard!

I found this ... problem ... in an old math book by a friend of my father's. However, there was no solution in the book, but I am curious about how this should be solved ...

And yes, the question was also written in the book like that, so without numbers or diagrams...

What is the context of the problem within that book? That is, is it just a list of random problems, or is there some topic the book is teaching, or at least something in common among nearby problems? I wouldn't call it "beginning algebra"; I can picture using a theorem from calculus, though that should not be necessary since everything here has constant speed.

I interpret the problem as saying that at some time two runners will both be at least 1/3 lap away from the other, in either direction. The word "lane" is odd, though. You did say you gave the exact wording, right?

I would tend to start by converting the problem to use relative speeds, by thinking of one runner as standing still, while the other two run at different speeds. We want to show that they will at some time both be within the opposite 1/3 of the track. (On the other hand, the problem doesn't indicate whether we can expect to be able to choose one runner ahead of time.)

 

[Thales12345]

What is the context of the problem within that book? That is, is it just a list of random problems, or is there some topic the book is teaching, or at least something in common among nearby problems? I wouldn't call it "beginning algebra"; I can picture using a theorem from calculus, though that should not be necessary since everything here has constant speed.

I interpret the problem as saying that at some time two runners will both be at least 1/3 lap away from the other, in either direction. The word "lane" is odd, though. You did say you gave the exact wording, right?

I would tend to start by converting the problem to use relative speeds, by thinking of one runner as standing still, while the other two run at different speeds. We want to show that they will at some time both be within the opposite 1/3 of the track. (On the other hand, the problem doesn't indicate whether we can expect to be able to choose one runner ahead of time.)

I'd have to check it out (I don't have the book with me), but I don't think there's a main topic. The word "lane" is literally in the book, but I also think they mean "lap". It is a fairly old book with some strange vocabulary at times, I also find the question a bit unclear.

(There was another question in the book I wrote down that I find interesting to discuss. It doesn't seem as vague as this one to me. But first this question! )

 

[Otis]

I found this proof

Hi Thales. You're not using the word 'proof' correctly. Once you demonstrate the claim, your work is the proof -- not the problem written in the book.

how this should be solved

You mean how it 'could' be solved. There exists a number of ways. (It should be solved correctly.)

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