Questions involving the pigeonhole principle in Combinatorial Analysis.

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Pigeonhole Principle about division

Prove that, for any $n+1$ integers $a_{1},a_{2},....,a_{n+1}$, there exist two of the integers $a_{i}$ and $a_{j}$ with $i \neq j$, such that $a_{i} - a_{j}$ is divisible by $n$. Please help me about ...
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let $A$ be a set of $n+1$ natural numbers between $1$ and $3n$. Show that there are $a,b \in A$ such that $n \leq a-b \leq 2n$

I'm having difficulties solving this question and would appreciate a nudge in the right direction. I think this is best solved with pigeonhole, but what are the pigeons and what are the holes?
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Pigeon Hole Principle (involving distances)

There are 100 old(non-digital) watches in an antique shop, all running but not necessarily on time. Prove that at some moment of time the sum of the distances from the center of the shop to the ...
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Pigeonhole principle problem involving circle and its chords

Several chords are drawn in a circle of radius 1 such that each diameter intersects no more than 4 of them. Prove that the sum of their lengths does not exceed 13.
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pigeonhole principle related problem

I'm given the problem: In a tournament which 18 teams participate, a team being matched with another in a round don’t match again in the follwoing (later) rounds. After 8 rounds prove that there are 3 ...
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Arangement of six circles in a plane

Six circles (including their circumferences and interiors) are arranged in the plane so that no one of them contains the center of another. Prove that they [the six circles] cannot have a point in ...
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650 points inside a circle of radius 16

There are 650 points inside a circle of radius 16. Prove there exists a ring with inner radius 2 and outer radius 3 covering 10 of these points. Hint of the professor: Use Dirichlet's (pigeonhole) ...
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Proof that Fibonacci Sequence modulo m is periodic?

It's well known that the Fibonacci sequence modulo m (where m is any integer) is periodic. I have figured out a proof for this, but upon googling, I found proofs online that were far more complicated. ...
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Pigeonhole Principle and maximum length of the repeating section

The question I have is, when 5 / 20483 is written as a decimal, what is the maximum length of the repeating section of the representation? I believe I need to divide 5 by 20483 which is equal to ...
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Small Combinatorical Question - Pigeonhole Principle Related

Suppose there are $77$ positive integers arranged in a row such that their sum is $140$. I want to show there is a sequence of adjacent integers in the row whose sum is $13$. My line of thought is ...
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Polygon and Pigeon Hole Principle Question

Seven vertices are chosen in each of two congruent regular 16-gons. Prove that these polygons can be placed one atop another in such a way that at least four chosen vertices of one polygon coincide ...
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pigeonhole principle with sequence of numbers

Let $(x_1,x_2,x_3,\dots,x_{77})$ be positive numbers. Use the pigeonhole principle to show that, if $\sum_{i=1}^{77}{x_{i}} = 140$, then there exist $j$ and $k$ such that $\sum_{i=j}^{k}{x_{i}} = 13$. ...
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Pigeon hole principle application

I am watching a lecture on pigeonhole principle at this link. At time 40:42, why does the instructor say that "either a will have 3 friends or 3 enemies". Why can't it be any of the other cases she ...
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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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How many people do you need to guarantee that two of them have the same initals?

An auditorium has a seating capacity of 800. How many seats must be occupied to guarantee that at least two people seated in the auditorium have the same first and last initials? I thought $26 \cdot ...
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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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Of any 52 integers, two can be found whose difference of squares is divisible by 100

Prove that of any 52 integers, two can always be found such that the difference of their squares is divisible by 100. I was thinking about using recurrence, but it seems like pigeonhole may also ...
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What is the minimum number that must share the same birthday (month & day) each year, given that one such birthday is February 29?

If there are 6,392 students at Stack Exchange College. What is the minimum number that must share the same birthday (month & day) each year, given that one such birthday is February 29?
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Show that any set of 7 distinct integers includes two integers $x$ and $y$, such that either $x-y$ or $x+y$ is divisible by 10

Show that any set of 7 distinct integers includes two integers $x$ and $y$, such that either $x-y$ or $x+y$ is divisible by 10. I'm trying to apply the pigeonhole principle, but haven't been able to ...
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Pigeonhole Principle: birthdays on same day of week

How many people must be in a room so that at least 10 have a birthday on a Friday? edit: Assume that no two people share the same birthday I'm somewhat confused and see two different ways to ...
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Pigeonhole Principle & Fermat's Little Theorm

I'm having a terrible time grasping Fermat's Little Theorem & then an even rougher time trying to use one to prove the other. Any help on this question would be tremendously appreciated! xx "The ...
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pigeonhole question with sets and sum of numbers

This question is meant to be solved with pigeonhole principle. But I can't solve it. I just can't figure out what is the pigeon and what is the pigeon hole. I don't really have a clear direction. ...
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Pigeonhole question - divisibility by chosen number

This question should be solved with pigeonhole principle. Let $a,n \in \mathbb N$ such that $a$ is a number whose digits are only $3$'s and $0$'s, and $n$ is an unspecified natural number. Show that ...
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Pigeonhole Principle question - sum of positive integers

A question that should be solved with pigeonhole but I'm having problems. $a_1,a_2,a_3,...,a_{77}$ are positive integers. We are given that $a_1+a_2+a_3+...+a_{76}+a_{77} < 133$ Show that there ...
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Picking three socks out of a drawer with two socks with two colors

How do I show that picking 3 socks containing just black and red socks that I must get either a pair of black or red socks? I mean it's fairly obvious, but how would I show it? Is this pigeon hole?
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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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Application of pigeonhole principle

Select $11$ diff erent numbers from $f\{1,2,...,20\}$. Prove that two of your numbers, $a$ and $b$, will diff er by two. Clearly this is an application of the pigeonhole principle. However, I'm not ...
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Generalized Pigeonhole Principle

Can somebody explain this to me? I am very confused. I have a question that says "What is the minimum number of students required in a discrete mathematics class to be sure that at least six will ...
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Pigeonhole problem - salvaging my solution

A student is solving combinatorics problems. Each day he solves at least one problem. He solves no more than 500 problems a year. Prove that there is an interval of days in which he solves 229 ...
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Max area of triangle -PHP

How do i prove that the maximum area that can be obtained among 3 random points in a square is half the area of the square?- I need it to for the following question " Show that among any 9 points ...
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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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Induction and typical pigeonhole principle

Let $n,\,k,\,r,\,s\in\mathbb{N}$ and $0\leq r,s<n$. We have $nk+r$ objects placed in $n$ containers. Show that we can choose $s$ containers such that there is at least $sk+\min{\{r,\,s\}}$ objects ...
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using the pigeon hole principle to prove that some integer with a sequence of ones and zeros is divisible by some d

Let $d$ be any fixed natural number. Show that there must exist an integer of the form $11\ldots1100\ldots 00$ (that is a integer whose digits consist of a sequence of $1$'s followed by $0$'s) which ...
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Pigeonhole Principle Question - Group of 6 people, do 3 either know each other or not?

Prove that in any group of 6 people there are always at least 3 people who either all know one-another or all are strangers to one-another. Hint: Use the pigeonhole principle. I don't see how this ...
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Prove that there is an element in the given set having fractional part less than 0.01

Given a set $ \{ \sqrt{3}, 2\sqrt{3}, 3\sqrt{3},...\}$, prove that some of the elements have fractional part less than 0.01 when written in decimal form. Here is my attempt so far: Divide the range ...
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Friend Group and Hater Group

Consider a set $S$ of $n$ people such that, for all distinct $x$ and $y$ in $S$, it is the case that either $x$ and $y$ like each other or $x$ and $y$ hate each other. Let us call $S' \subseteq S$ a ...
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pigeonhole principle divisibility proof

Let n be some positive odd number, prove that there exists some positive integer k such that n|(2k-1), prove in terms of the pigeonhole principle
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How do you tell which are the pigeons and which are the pigeon holes? [closed]

I am unable to correctly identify pigeons and pigeon holes in word problems. What is the technique?
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90 people with ten friends in the group. Prove its possible to have each person invite 3 people such that each knows at least two others

A high school has 90 alumni, each of whom has ten friends among the other alumni. Prove that each alumni can invite three people for lunch so that each of the four people at the lunch table will know ...
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Perfect Fourth Power - Pigeon Hole Principle

Let $a_1, a_2, ..., a_n$ be positive integers all of whose prime divisors are $\le$ 13. Show that if $n \ge 193$ then there exists four of these integers whose product is a perfect fourth power. I ...
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Proof Involving Pigeonhole Principle

Let $a_1, a_2, ..., a_n$ be positive integers all of whose prime divisors are $\le$ 13. a) Show that if $n \ge 65$ then there exist two of these integers whose product is a perfect square. [DONE] b) ...
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Pigeonhole Principle Proof

2004 flies are inside a cube of side 1. Show that some 3 of them are within a sphere of radius 1/11. I am not sure how to begin the proof especially since we are asked to work on a sphere rather than ...
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Extended Pigeonhole Principle: How to prove it?

A version of the pigeonhole principle is: (1) If m objects are put in n boxes and n < m, then at least one box contains at least ceil(m/n) objects An alternate (more generalized) version ...
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Prove that for positive number, some multiple only has 0 and d as it's digits

Let $ n$ be a positive integer, and let $1<=d<=9$. Show that some multiple of $n$ has $0$ and $d$ as its only digits. I don't know how to even start this question. It's under the pigeonhole ...
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With the pigeon hole principle how do you tell which are the pigeons and which are the holes?

For example, I was reading this example from my textbook: Let S be a set of six positive integers who maximum is at most 14. Show that the sums of the elements in all the nonempty subsets of S ...
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Pigeonhole principle problem involving inequality 0 < |$\sqrt{x} - \sqrt{y}$| < 1

21 integers are selected from {1, 2, 3, ..., 400}. Prove that two of them, say x and y, satisfy 0 < |$\sqrt{x} - \sqrt{y}$| < 1. I am confident you have to use and apply the Pigeon Hole ...
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Pigeonhole Principle / Number Theory

Let $S$ be a subset of $A=\{1,2,3,...,1000\}$. Find the largest number of elements in $S$ such that for any $a, b \in S$ with $a>b$, $a-b$ does not divide $a+b$. I've tried numerous approaches, ...
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Pigeonhole Question

This is an example from a Discrete math textbook: Any subset of size $6$ from the set $S = \{1,2, 4, \dots 9\}$ must contain two elements whose sum is $10$. Answer: Here the pigeons constitute a $6$ ...
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Pigeonhole question and generalization

Let H be a regular hexagon with side length 1 unit. (a) Show that if more than 6 points are speci ed inside H then the points of at least one pair of them are at most 1 unit apart. (b) State and ...
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Pigeonhole Principle Exercise

Show that any subset of $\{1, 2, 3, ..., 200\}$ having more than $100$ members must contain at least one pair of integers which add to $201$. I think it is doable using the Pigeonhole Principle.