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I was given the next question:

I need to prove that any subset with $25$ elements of the set $\{1,2,....150\}$, contain least $4$ elements, $x$, $y$, $z$, $t$ such that $x+y=z+t$.

Now my idea is to use the pigeon principle and set the pigeons to be the number of the possible pairings and the 'cages' to be the number of the possible sums of two elements from the set $\{1,2,....,150\}$ But I feel as if I'm missing something out and I'm not so sure about my answer. I'd like to hear your thoughts on this, thank you very much!

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Your idea is fine. We have ${25\choose2}=300$ pigeons and as the sums can only range from $1+2=3$ to $149+150=299$, there are only $297$ cages. Hence there are two different pigeons $\{x,y\}$ and $\{z,t\}$ in the same cage, i.e., $x+y=z+t$. Note that the pairs cannot overlap as for example $y=z$ would imply $x=t$ contradicting the distinctiveness of the pigeons.

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    Thank you very much! I feel like I'm getting better at this! And it is quite fun :) By the way, is there a different answer you'd give/think of? – 2017-01-07