Questions involving the pigeonhole principle in Combinatorial Analysis.

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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
37 views

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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0answers
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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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2answers
70 views

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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2answers
63 views

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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1answer
219 views

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
67 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 ...
2
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2answers
65 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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2answers
56 views

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
64 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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2answers
63 views

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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4answers
147 views

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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0answers
55 views

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
67 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". ...
5
votes
1answer
43 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
54 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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6
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3answers
528 views

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
42 views

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 ...
0
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1answer
28 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
147 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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0answers
41 views

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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0answers
35 views

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
32 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
79 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
119 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
26 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
70 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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0answers
75 views

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 $\epsilon_l = \pm 1$ ...
0
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1answer
35 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
99 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
29 views

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 ...
2
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2answers
77 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
311 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
123 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 ...
5
votes
2answers
105 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 ...
2
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2answers
77 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
64 views

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
50 views

$x_1,x_2,\ldots,x_m$ 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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0answers
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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
107 views

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
53 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
41 views

$[x-\frac{1}{n}, (n-1)x+\frac{1}{n}]$ 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
181 views

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
80 views

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
75 views

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 ...