# 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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### Proof related to pigeon hole principle to be done with induction

since the question is about a positive integer m, it's obvious that the use of mathematical induction needed, but to prove the fact for n = k+1 we have to use the pigeon hole principle, i am so ...
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### how to find pigeon holes in a question related to pigeon hole principle

prove that every set of 10 two digit numbers has two disjoint subsets with the same sum of elements. In this question i don't know how to choose the pigeon holes, or what will be the pigeon holes
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### Distance between points in a square [closed]

Let $p$ be a square whose side has length $1$. $51$ points are randomly chosen inside the square. Show that there are atleast $3$ points whose mutual distance is $< \sqrt{0.08}$.
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### Proving the Pigeonhole Principle

I am looking to prove the Pigeonhole Principle by proving the following claim: Let $A$ be a set with $m$ elements, and let $B$ be a set with $n$ elements, where $m,n\in \omega$ and $m > n$. ...
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### A group of $15$ boys plucked a total of $100$ apples. Prove that two of those boys plucked the same number of apples.

A group of $15$ boys plucked a total of $100$ apples. Prove that two of those boys plucked the same number of apples. My answer is: First distribute $90$ apples so that each will have $6$ apples. ...
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### Russia (2000) contest:Prove the existence of a pair of rows and columns with intersections differently coloured

We have a $100\times100$ board divided into $10^4$ unit squares. These squares are coloured with four colours so that every row and every column has $25$ squares of each colour. Prove that there are ...
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### Show that a class of nine students must have at least three male students or at least seven female students. [closed]

I am stuck with the following problem: Suppose that there are nine students in a class. Show that the class must have at least three male students or at least seven female students. Please help me ...
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### Applying pigeonhole principle to determine whether a list of strings must have duplicates.

Say you have a program that creates strings of lower-case letters of length 5 or less. It is told that the program holds 600,000 words on its drive. How can you figure out if all the words are ...