Questions involving the pigeonhole principle in Combinatorial Analysis.

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proving using pigeon hole principle

how would I prove this exercise: If we had five points in a square with sides of length one. How can we use the Pigeonhole Principle to prove that there are two of these points having distance at most ...
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1answer
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Dividing a triangle into seventeen equal parts.

I was trying to solve a problem on Pigeonhole principle from Problem Solving Strategies by Arthur Engel. A target has the form of an equilateral triangle with side 2 units. If it is hit ...
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Pigeon Hole Problem with 3 integers

So, given any set of three integers, prove there is a pair whose sum us even, and then prove or disprove that there is a pair whose sum is odd. To prove that there is a pair whose sum is even, ...
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Pigeon Hole Problem with coins

So, for this problem, I am told there is a jar containing: 13 pennies 12 nickels 9 dimes 8 quarters All of which are to be removed from the jar at random. So, ...
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pigeon hole principle and fermet little principle [on hold]

If a is the largest 21 digit multiple of 10^10+101 , b largest 21 digit multiple of 10^10+102 and c largest 21 digit multiple of 10^10+103 , output the following : if number =(3 * abc *(a^3 -b^3) * ...
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Qn on Pigeon-Hole Principle

Let S be a set of 10 positive integers ≤ 50. Show that there two different (but not necessarily disjoint) subsets of four integers such that the sums of the 4 integers in the sets are equal. Having ...
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2014 points inside a cube

$2014$ points are chosen inside a cube with side $13$. Can a cube with side $1$ be found inside it so that it doesn't contain any of chosen points? This must be a problem solved using ...
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2answers
687 views

Is there among first 100000001 Fibonacci numbers one that ends with 0000?

This is a difficult problem from competition training: Is there among first 100000001 Fibonacci numbers one that ends with 0000? Trainer says use pigeonhole principle. I do not know how.
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1answer
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If 51 mosquitoes are sitting on a square with side 1m, are at least 3 of them within a disk of radius 1/7?

There are 51 mosquitoes on a square-shaped window with side 1 m. Can Stephen kill 3 mosquitoes with a circular plastic disk of radius 1/7 m in a single strike? I know this can be solved by ...
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Pigeonhole Principle - set division

I'm unable to figure this problem out but my professor did suggest that it could be solved using Pigeonhole. Consider the set $X = \{1,2,3,4,5\}$ and suppose you have two holes. Also suppose you have ...
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Problem on pigeon hole principle

This is a problem based on pigeon hole principle. A tennis player has three weeks to prepare for a tennis tournament.She decides to play at least one set every day but not more than 36 in all.Show ...
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Logic Questions with Pigeonhole Principle

"If $n$ objects are distributed into $k$ boxes, then at least one box must contain at least _____ objects." Fill in the blank How many people must we have in a room to ensure that six of them were ...
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Pigeonhole Principle

Let $X = {x_0, x_1, · · · , x_m}$ be a subset of ${1, 2, · · · , n}$, where $m > n/2$, and $x_0$ is the smallest number in $X$. Use the pigeonhole principle to show that $X$ contains two numbers ...
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1answer
69 views

Pigeonhole proof of the existence of two numbers with given sum [duplicate]

Let $|W|=m+1$ and $W$ be a subset of $X=\{1,2,3,\dots ,2m\}$ ($m$ is any natural number). Prove there exists two numbers in $W$ whose sum is $2m+1$. Can anyone give me a hint to prove this? I ...
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2answers
69 views

Any two points inside a circle are within a diameter of each other.

In many problems involving the Pigeonhole Principle, we often assume the following lemma: Lemma: The distance between any two points in a circle of radius $r$ is at most $2r$. Intuitively, this ...
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Pigeonhole principle question - relatively prime

Prove that every subset A of the set {2, 3, ... 99, 100} with |A| > 26, has at least one pair of integers that is not relatively prime. 2, 3, 5 .. , there are 26 primes below 100. Can someone give ...
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1answer
71 views

Constructive proof of pigeonhole principle

I'm trying to prove Pigeonhole principle with Coq proof assistant. Here is how I defined it: ...
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Show that some 5 consecutive chairs must be occupied.

A group of 25 people are seated in a row of 30 chairs. Show that some 5 consecutive chairs must be occupied.
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How many subsets of $\{1, 2, …, n\}$ contain $1$ and how many don't? [closed]

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
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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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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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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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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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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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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 ...
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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
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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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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
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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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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
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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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1answer
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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
82 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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314 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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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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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
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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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$x_1,x_2,\ldots,x_m$ 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 ...