Questions involving the pigeonhole principle in Combinatorial Analysis.

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How many subsets of $\{1, 2, …, n\}$ contain $1$ and how many don't? [on hold]

Consider the set $A = \{1, 2, …, n\}$ (a) How many subsets of A contain $1$? I got $ 2^n - 2^{n-1}$ (b) How many subsets of A do not contain $1$? I got $2^{n-1}$ (c) Use the pigeonhole principle ...
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Elementary Pigeonhole Principle Question

Is my reasoning here correct? If not, advice would be appreciated. Thank you for your time! We assume that $A$ is finite and $f: A \rightarrow A$. We show that $f$ is one-to-one iff $ran \ f = A$. ...
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1answer
57 views

Number theoretical Application of the Pigeonhole Principle

I'm currently working through a paper related to my bachelors thesis and I'm stuck at a point where the author mentions the following result as "a standard application of the pigeonhole principle". ...
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1answer
34 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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1answer
49 views

Help prove $f:X \rightarrow Y$ is an injection $\Leftrightarrow$ $f:X\rightarrow Y$ is a surjection when $|X|=|Y|$

I need to prove: Given non-empty finite sets $X$ and $Y$ with $|X| = |Y|,$ a function $X\rightarrow Y$ is an injection if and only if it is a surjection. The hint given is to use the pigeonhole ...
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2answers
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3answers
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General Pigeonhole Principle - Coin Flips

I am trying to solve a problem using the general Pigeonhole Principle. The problem statement is as follows: A coin is flipped three times and the outcomes recorded. So, HTT might be recorded ...
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3answers
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123 persons in a cafe, and pigeons and boxes

$123$ persons are in a cafe. The sum of their ages is $3813$. Is it always possible to find $100$ among them so that their total age is greater or equal to $3100$? Looks like ...
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1answer
26 views

Regarding Thue's congruence theorem.

Did the mathematician Thue have a theorem where if $X\cdot N$ is congruent to $y \pmod m$, gcd$(y'm)=1$ then $1 \lt X \le \sqrt{m}$, or $1 \lt Y \le \sqrt{m}$? I'm not sure if I saw this in a number ...
4
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0answers
139 views

Select 100 integers from 1,2,…,200

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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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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1answer
30 views

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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2answers
78 views

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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2answers
106 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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1answer
25 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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1answer
64 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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1answer
30 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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2answers
69 views

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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1answer
94 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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1answer
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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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1answer
43 views

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
66 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
303 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
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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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2answers
86 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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2answers
75 views

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
47 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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1answer
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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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0answers
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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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2answers
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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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3answers
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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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2answers
49 views

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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1answer
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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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2answers
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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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1answer
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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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1answer
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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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1answer
54 views

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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1answer
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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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1answer
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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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3answers
64 views

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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1answer
255 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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3answers
160 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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1answer
127 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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2answers
103 views

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
141 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 ...