# Tagged Questions

Questions involving the pigeonhole principle, which states that if $n$ items are placed in $m$ containers and $n>m$, then one at least one container has more than one item.

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### Use Pigeonhole to show, of any set of $2^{n+1}-1$ positive integers is possible choose $2^{n}$ elements such that their sum is divisible by $2^{n}$.

Show that, of any set of $2^{n+1}-1$ positive integers numbers is possible choose $2^{n}$ elements such that their sum is divisible by $2^{n}$. My approach: Let $a_1,a_2,\ldots,a_{2^{n+1}-1}$ ...
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### A Combinatorics Problem with x+y = z

The numbers $\{1,2,...,2005\}$ are divided into $6$ disjoint subsets. Prove that for one of them we can find $x,y,z$ elements in it, not necessarily distinct such that $x + y = z$. I have no idea how ...
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### Show that there are always two teams who played exactly the same number of games.

So i was given this question. There are 11 teams in a league. Each team can play against the other team only once. Show that there are always two teams who played exactly the same number of games. My ...
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### 7 points on a closed disk, one must be the center

I am trying a pigeonhole strategy for proving this assertion of a MO test: "Given 7 points on a closed disk of radius 1 such that the distance between any two of this points is at least one, then one ...
### Choosing $7$ numbers from $[1,2,…,11]$ will give us $2$ that have sum $12$.
Choosing $7$ numbers from $[1,2,...,11]$ will give us $2$ that have sum $12$. I tried: There are only $5$ pairings possible: $(7,5),(8,4),(9,3),(10,2),(11,1)$ Suppose I pick $6$, and then not to be ...