Tagged Questions

3
votes
1answer
30 views

Smallest subset of $\{1,2,…,4n\}$ with a certain property

Fact 1: Let $A\subseteq\{1,2,...,2n\}$. If $n+1\leq |A|$, then there exists 2 elements $a,b\in A$ such that $a+b=2n+1$. Proof: This can be shown by writing $\{1,2,...,2n\}$ as the union of $n$ ...
0
votes
1answer
60 views

Combinatorics pigeonhole probems

Let there be $R$ red and $B$ blue balls, with each ball distinct from the other (even of the same colour). $M$ balls ($(1)$ assume $M<R,B$) are to be chosen. What is the probability that the number ...
0
votes
1answer
50 views

Ramsey's theory inequality with $t$-subsets

Let $q_{1},\, q_{2}, \ldots, q_{k},t$ be positive integers, where $q_{1}\geq t, q_{2}\geq t, \ldots, q_{k}\geq t$. Let $m$ be the largest of $q_{1},q_{2}, \ldots, q_{k}$. Show that ...
3
votes
3answers
128 views

Choose 38 different natural numbers less than 1000, Prove among these there exists at least two whose difference is at most 26.

Choose any 38 different natural numbers less than 1000. Prove that among the selected numbers there exists at least two whose difference is at most 26. I think I need to use pigeon hole principle, ...
4
votes
2answers
107 views

Guaranteeing an integer lattice point centroid

My question is this: Writing $n(4)$ to be the minimum number of integer lattice points in the plane so that some four of them must determine an integer lattice point centroid, show that $n(4)=13$. I ...
3
votes
1answer
97 views

pigeonhole principle 20 balls

I've worked out the answer to this as 13 since it's common sense, but we are supposed to apply the pigeon-hole principle, and I don't see how it is applicable here. A bowl contains 10 red balls ...
1
vote
2answers
48 views

Counting Subset Properties

Let $N=\{1,2,...,100\}$ and $A$ be a subset of $N$ with $|A|=55$. Show that $A$ contains two numbers with difference $9$. Is this also true for $|A|=54$? I was trying to solve this via the pigeonhole ...
3
votes
2answers
53 views

Pigeonhole Principle and Geometry

Consider any five points in the plane that have integer coordinates: -Prove that there are two points such that the midpoint of the line segment joining those two points also has integer coordinates
5
votes
1answer
96 views

Using Pigeonhole Principle to prove two numbers in a subset of $[2n]$ divide each other

Let $n$ be greater or equal to $1$, and let $S$ be an $(n+1)$-subset of $[2n]$. Prove that there exist two numbers in $S$ such that one divides the other. Any help is appreciated!
2
votes
1answer
66 views

Using the Pigeonhole Principle to show that $2$ of any $n+1$ numbers from $\{1,2,\ldots,2n\}$ sum to $2n+1$

Let n be greater or to 1, and let S be an (n+1)-subset of [2n]. Prove that there exist two numbers in S whose sum is 2n+1. I know I have to use the pigeonhole principle - no idea how to start...
11
votes
3answers
430 views

Arc sums for a circle of $k$ positive integers whose total sum is $n$

This problem got me thinking about the following more general scenario: Suppose you have $k$ positive integers with total sum $n$, and you arrange them in a circle. Given such an arrangement, you ...
6
votes
2answers
202 views

A Pigeonhole Principle problem

101 positive integers are placed on a circle whose sum is 300.Prove that it is possible to choose from these numbers some consecutive numbers whose sum is equal to 200. (I don't know if the word ...
1
vote
2answers
78 views

Given 33 natural number so that their prime divisor just with $ 7,5,2,3,11$.Prove that multiplication two number of these numbers are complete square

Given 33 natural number so that their prime divisor just with $ 7,5,2,3,11$ is formed. Prove that multiplication two number of these numbers are complete square. Thank you.
0
votes
1answer
125 views

Using the pigeonhole principle to prove there is at least two groups of people whose age sums are the same.

In a room there are 10 people, none of whom are older than 100 (ages are given in whole numbers only) but each of whom is at least 1 year old. Prove that one can always find two groups of people ...
0
votes
1answer
52 views

Another version of PP

Prove the following version of the pigeonhole principle. Let $m$ and $n$ be positive integers. If $m$ objects are distributed in some way among $n$ containers, then at least one container must hold at ...
0
votes
1answer
167 views

Three exercises related to the pigeonhole principle

I got three questions while writing some exercises. Questions (1) Suppose S is a set of 6 positive integers, whose maximum is 14. Prove that the sums of elements in all non-empty subsets of S ...
2
votes
1answer
110 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 ...
7
votes
2answers
280 views

Combinatorics proof

Suppose that $A$ is a set of 16 distinct natural numbers and that $1\leq p\leq100$ for every $p$ in $A $. Prove that $A$ contains 4 different numbers $a$, $b$, $c$, and $d$, such that $a+b=c+d$.
5
votes
3answers
182 views

combinatorics: The pigeonhole principle

Assume that in every group of 9 people, there are 3 in the same height. Prove that in a group of 25 people there are 7 in the same height. I started by defining: pigeonhole- heights. ...
2
votes
2answers
146 views

The decimal expansion of the quotient of two integers

It is an exercise in a book on discrete mathematics.How to prove that in the decimal expansion of the quotient of two integers, eventually some block of digits repeats. For example: $\frac { 1 }{ 6 } ...
1
vote
2answers
89 views

How are the pigeonholes calculated in this pigeon-hole problem?

The question is as follows: To prepare for a marathon, an elite runner runs at least once a day over the next 44 days, for a total of 70 runs in all. Show that there's a period of consecutive ...
53
votes
10answers
4k views

100 Soldiers riddle

One of my friends found this riddle. There are 100 soldiers. 85 lose a left leg, 80 lose a right leg, 75 lose a left arm, 70 lose a right arm. What is the minimum number of soldiers losing all ...
4
votes
1answer
486 views

Pigeonhole: Practical Applications in Computer Science

Most of the problems I've seen involving the pigeonhole principle have so far seemed fairly artificial. As I'm studying CompSci I'm interested what kind of practical, real world problems in CompSci ...
0
votes
2answers
297 views

Pigeonhole: 12 numbers between 10 to 100 - 2 have a difference divisible by 11

Prove that given 12 numbers between 10 to 100 - 2 have a difference divisible by 11. I didn't understand the answer given in my lecture and thought that as usual I'd probably get a clearer answer ...
3
votes
1answer
670 views

Combinatorics - pigeonhole principle question

This is for self-study. This question is from Rosen's "Discrete Mathematics And Its Applications", 6th edition. An arm wrestler is the champion for a period of 75 hours. (Here, by an hour, we mean a ...
6
votes
2answers
232 views

How to recognize a pigeonhole problem?

I'm going to split this into 2 questions, the first I think might have an answer, the second may not. First, is there a general way to recognize a pigeonhole problem as such? I mean are there some ...
1
vote
1answer
32 views

Min Number of Values from {1,2,…,9} Such that diff of 2 picked values is 5

This is a question from Shcaum's whose answer I don't understand. Our textbook has 2 pages on the pigeonhole principle and I'm having quite a bit of difficulty with it. Give the set ${1,2,...,9}$ ...
0
votes
1answer
52 views

Possibility of constructing a desirable subset

Here is a question.I am quoting it: Question by user Nahum Litvin Let A be a set of 100 natural numbers. prove that there is a set B B⊆A such that the sum of B's elements can be divided by ...
5
votes
2answers
211 views

regarding Pigeonhole principle

Let A be a set of 100 natural numbers. prove that there is a set B $$B\subseteq A$$ such that the sum of B's elements can be divided by 100 I am stuck for a few days now. Please help!
4
votes
1answer
188 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 ...
1
vote
2answers
281 views

Maximum number of mutually orthogonal latin square pairs (definition provided)

An $n\times n$ matrix is defined to be a "latin square" if each row and column is a permutation of the first $n$ natural numbers. Two squares of same order are orthogonal if the $n^2$ pairs ...
9
votes
2answers
1k views

In a group of 6 people either we have 3 mutual friends or 3 mutual enemies. In a room of n people?

A group of 6 people each pair is either a friend (acquaintance) or an enemy (stranger). It is to be proven that there are either 3 mutual friends or 3 mutual enemies in this group. I have an ad-hoc ...
3
votes
4answers
433 views

Another pigeonhole principle question

Have another question for you today: A course has seven elective topics, and students must complete exactly three of them in order to pass the course. If 200 students passed the course, show that at ...
8
votes
2answers
618 views

Chess Master Problem

From Introductory Combinatorics by Richard Brualdi We have a chess master. He has 11 weeks to prepare for a competition so he decides that he will practice everyday by playing at least 1 game a day. ...
1
vote
1answer
317 views

Pigeonhole principle question

Suppose a graph with 12 vertices is colored with exactly 5 colors. By the pigeonhole principle, each color appears on at least two vertices. True or false? The correct answer is false, but I assumed ...