Questions involving the pigeonhole principle in Combinatorial Analysis.

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four point in a row

We have painted all dots of page with two colors(blue and green), proof that there are four point with green color in a line that distance of any two neighbors of this four is one unit or there are ...
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A planar graph has either 2 faces or 2 vertices of degree less than 3

Practicing for an upcoming test, I stumbled upon this question: A planar graph with at least three vertices has either 2 faces of length at most 3, or 2 vertices of degree at most 3. Which is a ...
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Proving minimum exsistence of intersection cardinality

Let $F_1,F_2...F_{13}$ be sets such that $\forall 1\le i \le 13: F_i\subseteq [10]$ and $|F_i|=6$ when $[10]={1,2,3...10}$ prove that there are $1 \le j < k < l \le 13$ such that $|F_j \cap F_k ...
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Chords of a 20-gon

Twenty points lie on a circle, so as to form a regular polygon. Then they are split into ten pairs, and the points in each pair are connected by a chord. Prove that some pair of these chords have the ...
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124 views

Pigeonhole principle and finite sequences

Suppose we have $75$ boxes that are labeled from $1$ to $75$ and that in each box there is at least one ball, but there are not more than $125$ balls total. I'm trying to find the largest number $n ...
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37 views

10 non-increasing or non-decreasing sequence from 101 random numbers [duplicate]

In $101$ random integer numbers $a[i],i=0, \cdots,100$, prove that we can always find $10$ non-increasing or non-decreasing sequence. A sequence is a sequence of numbers is an array of numbers ...
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101 views

Combinatorics pigeonhole principle question.

There are $n$ people at a meeting, each of whom chooses $3$ distinct numbers between $1$ and $11.$ $\quad({\sf a})$ What is the smallest value of $n$ which guarantees that at least two people choose ...
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Pigeon Hole Principle in Unit Disk [duplicate]

Let $n$ be a natural number such that $n \ge 2$ and given complex number $z_1, z_2, \ldots, z_n$ that is contained in an open unit disk centered at origin. Prove that there exists $\epsilon_l = \pm 1$ ...
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41 views

Divisibility combinatorics

Let $A$ be a set of $1008$ positive integers bounded above by $2014$. It is then said that there must be two integers in $A$ such that one divides the other but I can't immediately see how to prove ...
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From any list of $131$ positive integers with prime factor at most $41$, $4$ can always be chosen such that their product is a perfect square

Author's note:I don't want the whole answer,but a guide as to how I should think about this problem. BdMO 2010 In a set of $131$ natural numbers, no number has a prime factor greater than 42. ...
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126 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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The Probabilistic Pigeon Hole Principle 2

(a) A group of 15 boys plucked a total of 100 apples. Prove that two of those boys plucked the same number of apples.
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Seems straightforward pigeonhole

If we are given $37$ integers then show that it is possible to choose $7$ of them with sum divisible by $7$. I have tried this problem but with no avail. If we assume there are no integers with ...
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2answers
136 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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2answers
395 views

Selecting from $\{1,2,3,4,5,6,7,8,9\}$ to guarantee at least one pair adds to $10$

How many numbers must be selected from the set $\{1,2,3,4,5,6,7,8,9\}$ to guarantee that at least one pair of these numbers add up to $10$? Justify your answer. Here's my answer. Consider the ...
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1answer
258 views

Pigeonhole Principle Question: Given any 5 points inside a square of side length 2, there is always a pair whose distance apart is at most $\sqrt2$

The question I am looking at: Prove that given 5 points inside a square of side length 2, it is always possible to find two of them whose distance apart is at most $\sqrt2$. This looks to me like I ...
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127 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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A question about the Pigeonhole Principle and linear equations over $\mathbb{Z}$

This may be a bit trivial (apologies if it is), but I was wondering if there was an elementary way to compute the cardinality of the solution set in the following situation: How many solutions would ...
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1answer
50 views

Show there exists a sequence of days within $49$ days where exactly $20$ hrs. are worked

Assume an integer number of hours will be worked each day for $49$ consecutive days. Further assume that at least $ 1 \frac{\text{hrs}}{\text{day}}$ and at most $11 \frac{\text{hrs}}{\text{wk}}$ can ...
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My first proof employing the pigeonhole principle / dirichlet's box principle - very simple theorem on real numbers. Please mark/grade.

What do you think about my first proof employing the pigeonhole principle? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Among three elements ...
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$x_1,x_2,\ldots,x_m$ is a permutation of $1,2,\ldots,m$ and $n_1,n_2,\ldots,n_m$ be integer and $m>1$ and odd.

$x_1,x_2,\ldots,x_m$ is a permutation of $1,2,\ldots,m$ and $n_1,n_2,\ldots,n_m$ be integer and $m>1$ and $m$ is odd. Now, $f(x)$ be defined as $f(x)=\sum\limits_{i,j=1}^{m} n_ix_j$. If $a,b$ are ...
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How do you justify the PigeonHole principle?

I am working on the problem below and just have two questions pertaining to my answers. 1) Am I clearly and correctly justfying my answers, anything I can improve on or explain better? 2) Are my ...
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Pigeon Hole Principle on a set of n elements

Homework question: It is asking us to prove that if we have $\frac{n}{2} + 1$ integers selected from a set$ A = {1, 2, ..., n}$, $n$ being an even integer, then the selection includes integers $x$ ...
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Show that for any subset of $10$ distinct integers there exists two disjoint subsets equal in sum

The title abbreviates the following homework exercise on the Pigeonhole Principle. Show that for any set of $10$ distinct integers from $1 \dots 60$ there exists two disjoint subsets of equal ...
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Prove:that in any set of 1009 positive integers exits two numbers $a_i$ and $a_j$ such that $a_i-a_j$ or $a_i+a_j$ is divisible by 2014

Show that in any set of 1009 positive integers exits two numbers $a_i$ and $a_j$ such that $a_i-a_j$ or $a_i+a_j$ is divisible by 2014 without remainder ($i\not=j$). I think the "pigeonholes" here ...
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$[x-\frac{1}{n}, (n-1)x+\frac{1}{n}]$ contains an integer $\forall x\in \mathbb{R}$ and $\forall n\in \mathbb{N}$

For any real number x: Prove that among the numbers x,2x,...,(n-1)x ,there is one that differs from an integer by at most $\frac{1}{n}$. any hints for a pigeon solution. Non-pigeon solution ...
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The Probabilistic Pigeon Hole Principle

Many people are aware of the Pigeonhole Principle: If we distribute $n+1$ pigeons into $n$ pigeonholes, at least one hole will contain at least two pigeons. However, much fewer are aware of the ...
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In a class of 50 students, how many students are guaranteed to get the same score on an equally-weighted 20 question quiz?

I am completely lost... Any help is greatly appreciated. I am unsure where to go to better understand the concepts behind this problem. The problem: In a class of 50 students, how many students are ...
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Probability Pigeonhole Principle

Choose any different 38 natural numbers less than 1000. Prove by using the Pigeonhole Principle that among the selected numbers there exists at least two whose difference is at most 26. I proved an ...
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pigeonhole principle question 40 participants in an art workshop

There are 40 participants in an art workshop. Each one of them signed up for one or more of the following courses: handicraft, ceramics and Chinese paintings. One of the combinations of courses must ...
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pigeons and pigeonhole [closed]

Twenty cards numbered 1 to 20 are placed face down on a table. Cards are selected one at a time and turned over. If two of the cards add up to 21, the player loses. Use pigeonhole principle to show ...
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Is there a Pigeon hole principle proof

Let $a_i$, $1 \leq i \leq 5$ denote five positive real numbers such that $\sum_{i =1}^{5}a_i = 100$. Show that there exist a pair $a_i,a_j$ such that $|a_i-a_j|\leq 10$. Is there a proof using pigeon ...
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Pigeonhole Principle and Equivalence Classes

Let $A$ be a finite set with $n \geq 4$ elements and let $\rho$ be an equivalence relation on $A$. Suppose that there are exactly four equivalence classes: $C_1, C_2, C_3, C_4$. Moreover we know that ...
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260 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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262 views

Pigeonhole Principle to Prove a Hamiltonian Graph

I am trying to figure out if a graph can be assumed Hamiltonian or not, or if it's indeterminable with minimal information: A graph has 17 vertices and 129 edges. ...
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138 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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How to estimate pigeonhole principle?

I was thinking about this after my professor mentioned the pigeonhole principle in class. Let's say we have $N$ items and $M$ containers. Here we assume $N > M$. We will randomly place each of the ...
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Pigeon Hole. 80 numbered balls

We have $80$ numbered balls(From $1$ to 80).Among which are $45$ blue and $35$ orange. Prove that at least two blue balls differ by $9$. For example $13$ and $22$ or $69$ and $78$. So they can differ ...
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Pigeon hole principle with sum of 5 integers

Prove that from 17 different integers you can always choose 5 so the sum will be divisible by 5. I tried with positive,negative numbers. Even, odd numbers etc but can't find the solution. Any ...
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4answers
151 views

Pigeonhole principle for dominoes.

Suppose we have 13 dominoes, each with a red and blue integer number. Prove that there is a subset of 4 dominoes such that the sum of the 4 red numbers and the sum of the 4 blue numbers are both ...
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Pigeon hole principle?

A guy reads a book. He read for 81 hours last 10 days. Prove that there has been two consecutive days when he read for at least 17 hours. 81 hours / 10 days equals 8,1 hour a day. 2 * 8,1 = 16,2. It ...
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Pigeonholing mod 4 points on plane.

I have the following problem as homework. Suppose there are 13 points in the plane, all with integer coordinates. Prove at least one quadrilateral with vertices on those points has a barycentre with ...
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2answers
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$2^n$ modulo n where n is odd always yields either an even or $1$

I'm attempting to do a pidgeonhole proof to prove that for some odd integer n, there is always a $2^k$ such that $2^k \mod(n) = 1$. I know that $2^n \mod(n)$ will always yield either an even number or ...
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pigeonhole principle For 1-1 and onto function

$|A|=|B|=n\in N$ Prove/Disprove that if $f:A \rightarrow B$ is 1-1 then $f:A \rightarrow B$ is onto. The answer I saw used the Pigeonhole Principle, I am trying to prove it without the principle. ...
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Pigeonhole Principle -

Let $n$ be an odd integer and let $f$ be an $n$-permutation of length $n$. Show that the number \begin{equation}x = (1-f(1))\cdot(2-f(2))\cdot...(n-f(n))\end{equation} is even. I don't understand ...
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1answer
80 views

Proving something using Pigeonhole Principle [duplicate]

How do I prove the following using the Pigeonhole principle? Let $n$ be an odd integer. Prove that there exists a positive integer $k$ such that $2^k \mod n = 1$. I don't understand how I can prove ...
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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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if there are 5 points on a sphere then 4 of them belong to a half-sphere.

If there are 5 points on the surface of a sphere then there is a closed half sphere containing at least 4 of them. It's in a pidgeonhole list of problems, but I think I have to use rotations in more ...
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If n is an odd integer, show there exists a positive integer k such that 2^k mod n = 1.

Hi I've been trying to solve this problem for at least 4 hours now but I can't figure it out. If anyone can help I would really appreciate it! I am asked to prove this using the pigeonhole principle: ...
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64 views

If $n\nmid a,a+d,a+2d. . . a+(n-1)d$,then $(n,d)=1$

None of the numbers in the sequence $a,a+d,a+2d,a+3d. . . a+(n-1)d$ are divisible by $n$.Then we have to prove that d and n are coprime. I am supposed to use the pigeonhole principle for this ...