Thread: Set of 13 real numbers

Hi, I'm having difficulty getting started with this problem:

Show that given any set of 13 distinct real numbers, there will be two elements in the set (x and y) such that


I really don't know where to start, but this is a course in combinatorics, and the pigeonhole principle is supposed to be used somewhere (probably in the part about sets)

1) How does the set having 13 elements come into play? I know that if we're talking about integers, then it would probably be related to some modulo thing (forcing 2 elements to have the same residue) but we're talking about real numbers

2) What is so special about the expression ? I mean, is this an expression of a property? Like distance formula or something? The closest thing I can get to is the difference of inverses (i.e. , but I don't know where the "1+" part is coming from.

3) What's the significance of ? Could it be replaced by anything else?

Basically, I get what's important, but I don't get what's connecting them ... any help with those connections would be great

Thanks in advance!

Originally Posted by Bingk

Hi, I'm having difficulty getting started with this problem:

Show that given any set of 13 distinct real numbers, there will be two elements in the set (x and y) such that


I really don't know where to start, but this is a course in combinatorics, and the pigeonhole principle is supposed to be used somewhere (probably in the part about sets)

1) How does the set having 13 elements come into play? I know that if we're talking about integers, then it would probably be related to some modulo thing (forcing 2 elements to have the same residue) but we're talking about real numbers

2) What is so special about the expression ? I mean, is this an expression of a property? Like distance formula or something? The closest thing I can get to is the difference of inverses (i.e. , but I don't know where the "1+" part is coming from.

3) What's the significance of ? Could it be replaced by anything else?

Basically, I get what's important, but I don't get what's connecting them ... any help with those connections would be great

Thanks in advance!

Hint(I think):




Define , for all and use Dirichlet's principle.

Thanks! I didn't think about using the tangent function, but everything came together with that