2
votes
2answers
101 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 ...
2
votes
2answers
54 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 ...
-4
votes
1answer
53 views

can any one help me with this pigeonhole question?

I try to solve it but i don't have enough time cuz it dues tomorrow. And I have no clue to solve it. hope you guy can help me out with this!
1
vote
2answers
53 views

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 $x$ ...
1
vote
3answers
84 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 ...
2
votes
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 ...
1
vote
1answer
57 views

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 ...
1
vote
2answers
916 views

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: ...
4
votes
1answer
57 views

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?
1
vote
1answer
47 views

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 ...
0
votes
1answer
104 views

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) ...
0
votes
2answers
118 views

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 ...
3
votes
1answer
61 views

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 ...
1
vote
1answer
72 views

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. ...
1
vote
1answer
47 views

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 ...
1
vote
1answer
73 views

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 ...
0
votes
3answers
61 views

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?
3
votes
3answers
117 views

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' ...
0
votes
1answer
61 views

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 ...
0
votes
2answers
171 views

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 ...
1
vote
0answers
132 views

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 ...
0
votes
1answer
81 views

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 ...
1
vote
3answers
117 views

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) ...
1
vote
2answers
103 views

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 ...
5
votes
5answers
120 views

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 ...
0
votes
1answer
62 views

Need help to prove pigeonhole problem

If we pick n+1 different positive integers with every integer is less than 2n. Prove that we can always find three numbers among these n+1 numbers that one is equal to the sum of the other two ...
0
votes
1answer
89 views

Pigeonhole principle questions

I want to solve the following problems with Pigeonhole principle. Show that in every group of people that have atleast 2 people, we can find couple that know the number of the people in the group.( ...
1
vote
1answer
42 views

No. of functions satisfying a certain condition

This is from an old exam: Let $M$ be a set of functions from $\mathbb{Z}/3$ into itself. What is the least number of elements that $M$ must contain for there to surely be at least two elements ...
2
votes
1answer
56 views

Prove that there exists a numbered socket such that for every orientation, two equal numbers coincide

There is a socket which has $6$ holes on the vertices of a regular hexagon. These holes are numbered $1, 2, \dots , 6$. Prove that there exists such a plug with $6$ prongs numbered such that no matter ...
4
votes
1answer
1k views

Prove that at a party with at least two people, there are two people who know the same number of people…

Okay, now, I really want to solve this on my own, and I believe I have the basic idea, I'm just not sure how to put it as an answer on the homework. The problem in full: "Prove that at a party ...
2
votes
4answers
503 views

Pigeonhole principle: show that a class of nine has at least five male or five female students.

Here is the problem in full, start to finish, with no other special instructions or rules: "If there are 9 students in a class, show that at least 5 must be male or at least 5 must be female. Also, ...
3
votes
2answers
107 views

Pigeonhole principle and a decagon

This is a homework Question and has to do with Pigeonhole principle. Could use a hint. Q. The numbers ${0,1,2,.....9}$ are randomly assigned to the vertices ${x_0,x_1,...x_9}$ of a decagon. Show that ...
1
vote
2answers
120 views

pigeonhole principle homework question

These are Homework question. They are pigeon hole principle questions and I have a very hard time with these unless I have worked on a similar problem before. Q.1. Prove that if we select 87 numbers ...
0
votes
1answer
82 views

Pigeonhole Principle Question…Fifteen different integers from 100 to 199 are given.

Question was too long to fit on title. Fifteen different integers from 100 to 199 are given. Show that it is always possible to select from these 15 integers at least two different sets $\{a_1, ...
1
vote
3answers
95 views

Pigeonhole principle application sums and differences

Let $A \subset \{1,2,...,99\}$, prove or disprove the following: a. For $|A| = 27$ b. For $|A| = 26$ There are $2$ different numbers in $A$ that their sum or their difference can be divided with ...
0
votes
1answer
67 views

There are 50 rooms in a line. If there are 26 rooms with girls, prove there are two girls exactly 5 rooms apart.

There are 50 rooms in a line. If there are 26 rooms with girls, prove there are two girls exactly 5 rooms apart. My idea was place 25 girls in into pairs of rooms, and there is no scenario which ...
0
votes
3answers
76 views

Pigeon holes principle

Let $P$ be a group that it's elements are 257 sentences in which only atomic sentences from $A,B,C$ exist (i.e. $A \iff B,\space\space A \wedge B \wedge C, \space\space...$) Show that there exists two ...
3
votes
1answer
101 views

Pigeonhole proof of Rational Approximation Theorem

I am stuck with the solution to the following problem (it is also known as the Rational Approximation Theorem) at the Art of Problem Solving wiki, which states: Show that for any irrational $x \in ...
0
votes
1answer
60 views

Using The Pigeon-Hole Principle

Let n be a positive integer. Show that in any set of n consecutive integers there is exactly one divisible by n. Here is the solution: Let $a,~a+1,...,a+n-1$ be the integers in the sequence. ...
0
votes
4answers
308 views

Can you help me solve these questions related to a Logical theory?

In a group of 200 people, number of people having at least primary education (assuming - Category I): number of people having at least middle school education (Category II): number of people having ...
2
votes
2answers
143 views

Pigeonhole-principle with two choices

I am able to solve this sort of problem pretty easily. An arm wrestler is the champion for a period of 75 hours. The arm wrestler had at least one match an hour, but no more than 125 total ...
1
vote
3answers
292 views

Pigeon hole birthday problem?

If there are 10,000 people, how many people must have the same birthday (ignoring year)? This is the way I went about this problem: 10000 people / 365 days in a year = 27.397 people per day ...
3
votes
1answer
318 views

Divisibility and Pigeonhole principle

Given a sequence of $p$ integers $a_1, a_2, \ldots, a_p$, show that there exist consecutive terms in the sequence whose sum is divisible by $p$. That is, show that there are $i$ and $j$, with $1 \leq ...
3
votes
3answers
150 views

Proof using pigeonhole and greatest integer (floor) function.

The question is to prove that if m is a positive integer then, $$[mx] = [x] + \left[x+\frac{1}{m}\right] +\left[x+\frac{2}{m}\right] + \cdots + \left[x+\frac{(m-1)}{m}\right] $$ for $x \in ...
2
votes
2answers
300 views

Pigeonhole Principle Points in a Triangle

Suppose we have an equilateral triangle with side length $1$. In this equilateral triangle, we place $8$ points either on the boundary or inside the triangle itself. Then what is the maximum possible ...
4
votes
2answers
832 views

Some three consecutive numbers sum to at least $32$

Here's a question we got for homework: We write down all the numbers from $1$ to $20$ in a circle. Prove that there is a sequence of $3$ numbers whose sum is at least $32$. I assume we need the ...
4
votes
1answer
218 views

$16$ natural numbers from $0$ to $9$, and square numbers: how to use the pigeonhole principle?

There are $16$ natural numbers placed next to each other. Each is a number from $0$ to $9$. These are in any order, and you can have as many repeats as you want (e.g. all $16$ numbers can be zero, or ...