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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In a sequence of $n$ integers, must there be a contiguous subsequence that sums to a multiple of $n$?

Let $x_1, \ldots, x_n$ be integers. Then are there indices $1\le a\le b\le n$ such that $$\sum_{i=a}^b x_i$$ is a multiple of $n$?
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3answers
182 views

Pigeonhole Principle Problem combo inequality

Prove that for any subset of $\{1,2,3,...,300\}$ with $102$ elements, there exists elements $M$ and $x$ in that subset such that $100<M-x<200$. I think this is a pigeonhole problem, I wanna ...
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3answers
454 views

combinatorics: The pigeonhole principle

Assume that in every group of 9 people, there are 3 in the same height. Prove that in a group of 25 people there are 7 in the same height. I started by defining: pigeonhole- heights. ...
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1answer
85 views

On a strange pigeonhole principle problem

Given distinct integers $a_1, a_2, \cdots, a_{63}$. Prove that there exists $a_i, a_j, a_m, a_n$ such that $(a_i - a_j)(a_m - a_n)$ is divisible by $1984$. I have no idea of how to create the ...
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2answers
217 views

Collection of numbers always in increasing or decreasing order

Anyone have any ideas on this question? I think you have to use the pigeon hole principle..but I am not sure about that? The numbers $1,2,3,\ldots,101$ are written down in a row in some order. Is ...
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1answer
4k views

Prove that at a party with at least two people, there are two people who know the same number of people…

Okay, now, I really want to solve this on my own, and I believe I have the basic idea, I'm just not sure how to put it as an answer on the homework. The problem in full: "Prove that at a party ...
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1answer
92 views

Putnam Problem, Pigeonhole Principle

I have never attempted or considered any contest math problems, but I recently found a page of Putnam Prep problems in a recycling bin on campus and decided to give some a try since I am home for ...
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1answer
113 views

Problem with $20$ integers less than $70$

20 pairwise distinct integers each less than 70 are taken and their pairwise differences are taken(magnitude of the difference). Show that there always exists 4 equal numbers. I somehow found ...
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2answers
343 views

Pigeonhole Principle and Sets

Can anyone point me in the right direction for this homework question? I know what the pigeonhole principle is but don't see how it helps :( Let $n\geqslant 1$ be an integer and consider the set S = ...
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1answer
2k views

Combinatorics - pigeonhole principle question

This is for self-study. This question is from Rosen's "Discrete Mathematics And Its Applications", 6th edition. An arm wrestler is the champion for a period of 75 hours. (Here, by an hour, we mean a ...
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1answer
186 views

Set theory: Metamath Proof of the Pigeon-Hole Principle, Error?

I have recently came discovered Metamath. Supposedly the language is one that a computer may proof-check. I then began to look at concepts that I am familiar with, and decided to look up the pigeon ...
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Improving statement obtained by Pigeonhole principle

In this MSE question, this statement is proven: Room is cube-shaped, with side 3m. 136 flies fly in it. Prove that at any moment one can encompass 6 flies with a sphere of radius 90cm. Can ...
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2answers
1k views

Existance of multiple of $n$ with only 0 and 1 as it's digits [duplicate]

Possible Duplicate: Proof that a natural number multiplied by some integer results in a number with only one and zero as digits I read this somewhere recently: For any natural number $n$, ...
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2answers
2k views

Some three consecutive numbers sum to at least $32$

Here's a question we got for homework: We write down all the numbers from $1$ to $20$ in a circle. Prove that there is a sequence of $3$ numbers whose sum is at least $32$. I assume we need the ...
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4answers
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Another pigeonhole principle question

Have another question for you today: A course has seven elective topics, and students must complete exactly three of them in order to pass the course. If 200 students passed the course, show that at ...
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3answers
190 views

Pigeonhole Principle Homework Problem

Seven boys and five girls are seated (in an equally spaced fashion) around a circular table with 12 chairs. Prove that there are two boys sitting opposite one another. I used 'G' for girls and 'B' ...
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4answers
456 views

How to draw Square Diagonal? [duplicate]

Draw a 5x5 square. In 16 of 25 squares draw diagonals in such a way that no diagonal ends touch. How can I do this?
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1answer
281 views

discrete math about Pigeonhole Principle

Prove that any set of $10$ positive integers less than or equal to $100$ will always contain two subsets with the same sum. Can anyone help me with this problem? Thanks.
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3answers
394 views

pidgeonhole problem need assistance

Suppose you have a sequence 2014, 20142014, 201420142014, . . . Show that there is an element in this sequence such that it is divisible by 2013. This is a problem I had on an exam and I know that ...
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1answer
177 views

Math and chess question!

Given a $6\times6$ chess board with $13$ marked squares, can you always place three mutually non-attacking rooks on the marked squares? If so, how can this be proven?
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2answers
168 views

Minimum number of coins to ensure 10 coins of one type are selected

One coin is labelled with the number $1$, two different coins are labelled with the number $2$, three different coins are labelled with the number $3$, $\ldots$ , forty-nine different coins are ...
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4answers
117 views

In every set of $14$ integers there are two that their difference is divisible by $13$

Prove that in every set of $14$ integers there are two that their difference is divisible by $13$ The proof goes like this, there are $13$ remainders by dividing by $13$, there are $14$ numbers ...
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1answer
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Pigeonhole: Practical Applications in Computer Science

Most of the problems I've seen involving the pigeonhole principle have so far seemed fairly artificial. As I'm studying CompSci I'm interested what kind of practical, real world problems in CompSci ...
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3answers
162 views

Application of the pigeon hole principle

There's a problem in my textbook that I'm having difficulties understanding. The solution given skips steps and is hard to decipher. The problem is as follows: "During a month with 30 days, a ...
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2answers
83 views

Pigeonhole principle on two coloured circle

Suppose a circle is divided into 200 congruent sectors, with 100 of them coloured red and the other 100 blue. A smaller concentric circle is placed on the larger circle and also so divided and ...
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2answers
283 views

Combinatorics Pigeonhole problem

Hello to all! So i have to do this problem: In the course of an year of 365 days Peter solves combinatorics problems. Each day he solves at least 1 problem, but no more than 500 for the year. Prove ...
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2answers
309 views

Guaranteeing an integer lattice point centroid

My question is this: Writing $n(4)$ to be the minimum number of integer lattice points in the plane so that some four of them must determine an integer lattice point centroid, show that $n(4)=13$. I ...
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2answers
411 views

Prove two numbers of a set will evenly divide the other

We have a set A of numbers 1, 2, 3... to 200 The question is asking me to prove that if I choose 101 numbers from the set, there will be two such that one evenly divides the other. I know this ...
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3answers
184 views

proof using pigeonhole principle

I am struggling to come up with a proof to the following question(from cut-the-knot.org): Prove that if n is odd,then for any permutation $p$ of the set $\{1,2,3...,n\}$ the product $$P(p) = ...
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1answer
265 views

$16$ natural numbers from $0$ to $9$, and square numbers: how to use the pigeonhole principle?

There are $16$ natural numbers placed next to each other. Each is a number from $0$ to $9$. These are in any order, and you can have as many repeats as you want (e.g. all $16$ numbers can be zero, or ...
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1answer
159 views

Students knowing others

There are 25 students in the class. It is known that among any three of them, two know each other. Show that there is a person who knows at least 12 other people. Thoughts: I know this is true since ...
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1answer
282 views

10 points inside a square - minimum distance between any of them

A square of side 1 is given, and 10 points are inside the square. If we divide the square into 9 smaller squares, and apply Dirichlet principle, we can prove that there are 2 of these 10 points whose ...
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2answers
134 views

If ten points are on a unit square, one pair is at most $\sqrt2/3$ apart

Ten points are placed in a unit square. Show that there is a pair of points at most $\sqrt2/3$ apart. I'm not sure how to proceed with this problem, and have not had any luck so far.
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1answer
676 views

Divisibility and Pigeonhole principle

Given a sequence of $p$ integers $a_1, a_2, \ldots, a_p$, show that there exist consecutive terms in the sequence whose sum is divisible by $p$. That is, show that there are $i$ and $j$, with $1 \leq ...
4
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1answer
350 views

If any x points are elected out of a unit square, then some two of them are no farther than how many units apart?

If 5 points are randomly positioned in a unit square, no two points can be greater than square root of 2 divided by 2 apart; divide up the unit square into four squares, and, based on the pigeonhole ...
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1answer
52 views

Prove using induction that from a set of $n+1$ numbers from $1..2n$, at least one number will evenly divide another.

Given a set of $n+1$ numbers out of the first $2n$ natural numbers, $1,2,\ldots,2n$, prove that there are two numbers in the set, one of which divides the other. I can't tell if I'm reducing the ...
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3answers
84 views

On the distribution of multiples of 7 into intervals of length 11

Say we have two primes, say 7 and 11. We are to consider the positions of the multiples of 7 inside the (7 buckets of) multiples of $11$. So the buckets of 11 are: $[1,11],[12,22],\ldots ,[67,77]$, ...
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4answers
226 views

Pigeonhole principle (I think): colored points in the plane

Suppose that each points in $\Bbb R^2$ is colored red, green or blue. Prove that either there are two points of the same color a distance $1$ unit apart, or there is an equilateral triangle of side ...
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1answer
97 views

Elegant proof of icosohedron property

This problem was question A1 on the 2013 Putnam contest. Is there a better way to solve this problem than just using pigeonhole principle? Specifically, is there a group theoretic way to interpret ...
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1answer
266 views

Issue concerning enumerating vertices in a prism (number of two adjacent vertices can only differ by a certain amount)

There are 100 vertices in a prism with a 50-gon as its base. Those vertices are assigned integers 1 to 100 (inclusive) in a random order. Each number can only be assigned once. The objective is to ...
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1answer
157 views

If one eats $100$ chocolates in $58$ days,then he must be eating exactly 15 chocolates in some consecutive days

BdMO 2014 Nationals $X$ eats 100 chocolates in 58 days,eating at least 1 chocolate per day.Prove that,in some consecutive days,she ate exactly 15 chocolates. I tried using the pigeonhole ...
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1answer
322 views

Ramsey Type problem (variant of people at a party)

There is $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\lfloor n/2\rfloor-1$ of them, each of whom either knows both or else ...
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0answers
187 views

Select 100 integers from 1,2,…,200 [duplicate]

Prove that if 100 integers are chosen from 1,2,...,200, and one of the integers chosen is less than 16, then there are two chosen numbers such that one of them is divisible by the other. Thanks in ...
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3answers
790 views

A problem in discrete math (Pigeonhole principle related)

This question is from the multiple choice test: A circular region is divided by 5 radii into sectors, as shown above. Twenty-one points are chosen in the circular region, none of which is on any of ...
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2k views

Subsets with equal sums

I have a problem to solve but I am in need of your help. Subjects with equal sums: Prove that for every set $A$ which consists of $10$ double digit natural numbers( numbers among $10, \ldots, 99$), ...
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The Pigeon Hole Principle and the Finite Subgroup Test

I am currently reading this document and am stuck on Theorem 3.3 on page 11: Let $H$ be a nonempty finite subset of a group $G$. Then $H$ is a subgroup of $G$ if $H$ is closed under the ...
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2answers
160 views

Pigeonhole Principle - numbers between $1$ and $100$

Of the set $A=${$1,2,...,100$}, we will choose $51$ numbers. Prove that, among the $51$ chosen numbers, there are two such that one is multiple of the other My notes: 1) There are $25$ prime numbers ...
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1answer
208 views

Arrangement of $100$ points inside $13\times18$ rectangle

Prove that you can't arrange 100 points inside a $13\times18$ rectangle so that the distance between any two points is at least 2. I tried many ways to divide the rectangle, but I can't get the ...
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1answer
84 views

Prove by using Pigeon Hole Principle

Let $k \in \mathbb Z^+ $. Prove that there exists a positive integer $n $ such that $k|n$ and the only digits in $n$ are 0's and 3's
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4answers
120 views

Let $S=\{3,4,5,6,7,8,9,10,11,12\}$. Suppose 6 integers are chosen from S. Must there be 2 integers whose sum is 15?

How can I go about solving this Pigeonhole Principle problem? So I think the possible numbers would be: $[3+12], [4+11], [5+10], [6+9], [7+8]$ I am trying to put this in words...