Questions involving the pigeonhole principle in Combinatorial Analysis.

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pigeonhole principle - 100 points in $13\times18$ rectangle

Prove that you can't arrange 100 points inside a $13\times18$ rectangle so that the distance between any two points is at least 2. I tried many ways to divide the rectangle, but I can't get the parts ...
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127 views

An interesting problem using Pigeonhole principle

I saw this problem: Let $A \subset \{1,2,3,\cdots,2n\}$. Also, $|A|=n+1$. Show that There exist $a,b \in A$ with $a \neq b$ and $a$ and $b$ is coprime. I proved this one very easily by using pigeon ...
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123 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...
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162 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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Prove, that in the subset of $\{1,\ldots,150\}$ there are two disjoint pairs with the same sums.

Prove, that in the subset with cardinality $25$ of $\{1,\ldots,150\}$ there are two disjoint pairs with the same sums. Well, there are at most $150+149=299$ possibilities of sums. But if we have a ...
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Extended Pigeonhole Principle: How to prove it?

A version of the pigeonhole principle is: (1) If m objects are put in n boxes and n < m, then at least one box contains at least ceil(m/n) objects An alternate (more generalized) version ...
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A combinatorics problem

Given $A = \{a_0, a_1,...,a_m\}$ such that it's a subset of $\{1,2,...,n\}$ where $m>n/2$, and $a_0$ is the smallest number in $A$. Show that $A$ contains two numbers $b$ and $c$ such that ...
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396 views

Pigeonhole Principle on Graphs

I just have a last minute question for my combinatorics final (which is in one hour!!). My prof particularly told me to study the following question and I'm pretty sure it involves the pigeonhole ...
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145 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 ...
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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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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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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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70 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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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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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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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 ...
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$2^n$ modulo n where n is odd always yields either an even or $1$

I'm attempting to do a pidgeonhole proof to prove that for some odd integer n, there is always a $2^k$ such that $2^k \mod(n) = 1$. I know that $2^n \mod(n)$ will always yield either an even number or ...
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Prove that in each coloring of a $4\times7$ board in two colors there's a square that all four of it's corners are colored by the same color

Prove that in each coloring of a $4\times7$ board in two colors there's a square that all four of it's corners are colored by the same color. This is a pigeon hole principle question and I have a ...
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1answer
77 views

Help with a pigeonhole principle?

Let $n \geq 1$ be an integer. Use the Pigeonhole Principle to prove that in any set of $n + 1$ integers from $\{1, 2, . . . , 2n\}$, there are two integers that are consecutive (i.e., di ffer by ...
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pigeonhole principle with sequence of numbers

Let $(x_1,x_2,x_3,\dots,x_{77})$ be positive numbers. Use the pigeonhole principle to show that, if $\sum_{i=1}^{77}{x_{i}} = 140$, then there exist $j$ and $k$ such that $\sum_{i=j}^{k}{x_{i}} = 13$. ...
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Application of pigeonhole principle

Select $11$ diff erent numbers from $f\{1,2,...,20\}$. Prove that two of your numbers, $a$ and $b$, will diff er by two. Clearly this is an application of the pigeonhole principle. However, I'm not ...
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pigeonhole principle divisibility proof

Let n be some positive odd number, prove that there exists some positive integer k such that n|(2k-1), prove in terms of the pigeonhole principle
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Probability problem- Are there atleast 3 balls with same colour radius and in same box?

The question is as follows, it came in RMO in 1990. It most likely involves probability..as far I think. Two boxes contain between them 65 balls of several different sizes. Each ball is white, black, ...
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A game involving points in the integer plane - who wins?

I am running a workshop on puzzles and problem solving over the weekend and thought that it might be a good idea to get people engaged by phrasing some interesting mathematical results in terms of ...
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A family of $n$ non-zero vectors of an $(n-1)$-dimensional vector space must be linearly dependent

I was bored earlier and began to think of the pigeonhole principle, and it came to me that it could be used to show that a family of $n$ non-zero vectors of an $(n-1)$-dimensional vector space must be ...
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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 ...
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256 views

Pigeonhole principle to prove division

Here's a little question that we were shown in class: Let $S = \{1,2,\ldots,200\}$ and let $A \subseteq S$ such that $|A| = 101$. Prove that there are two elements of $A$ such that one is a ...
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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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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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Proving the Chinese Remainder Theorem using the Pigeonhole Principle

I am trying to prove a version of the Chinese Remainder Thoerem using the pigeonhole principle. The theorem that was provided: If n and m are relatively prime, then for all integers 0 ≤ a < n ...
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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 ...
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1answer
50 views

Average number of pigeon holes.

I'm an engineer not a mathematician, and I have a 3 part question that's applicable to a parallel computer system my team is designing. We have 10 CPU cores (ie - 10 pigeons) randomly reading from 10 ...
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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 ...
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Prove that there is an element in the given set having fractional part less than 0.01

Given a set $ \{ \sqrt{3}, 2\sqrt{3}, 3\sqrt{3},...\}$, prove that some of the elements have fractional part less than 0.01 when written in decimal form. Here is my attempt so far: Divide the range ...
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With the pigeon hole principle how do you tell which are the pigeons and which are the holes?

For example, I was reading this example from my textbook: Let S be a set of six positive integers who maximum is at most 14. Show that the sums of the elements in all the nonempty subsets of S ...
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Discrete Mathematics - Ice Cream random samples

How would you solve the following problem with Discrete Mathematics, and what is the answer? Suppose there are 5 different types of ice cream you like. How many random samples ice cream must be eaten ...
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Difference of two powers of $3$ divisible by $2011$

How to prove that there exists two powers of $3$ that differ by a number that is divisible by $2011$?
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How to apply pigeonhole principle to this problem?

There are 33 students in the class and sum of their ages 430 years. Is it true that one can find 20 students in the class such that sum of their ages greater 260 ? My approach: The average age of ...
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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 ...
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Pigeonhole principle problem

The problem I'm working on says: A basketball player has been training for 112 hours during 12 days. He has trained an integer number of hours every day. Prove that there was two consecutive days ...
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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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using the pigeon hole principle to prove that some integer with a sequence of ones and zeros is divisible by some d

Let $d$ be any fixed natural number. Show that there must exist an integer of the form $11\ldots1100\ldots 00$ (that is a integer whose digits consist of a sequence of $1$'s followed by $0$'s) which ...
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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) ...
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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 ...
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Pigeon principle question: Nine points in a diamond

A diamond (a parallelogram with equal sides) is given, and its sides are 2 cm long. The sharp angels are 60 degrees. If there are nine points inside the diamond, prove that there must be two of them ...
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Bit strings (pigeonhole principle)

Here is how the question is posed: Let $s_1$, $s_2$, $s_3, \ldots, s_{90}$ be 90 bit strings of length nine or less. Prove that there exist two strings $s_i$ and $s_j$ with $i \neq j$ that contain ...
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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 ...
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using pigeonhole principle for a hand of thirteen cards

Say I shuffle and deal a hand of thirteen cards. How can I apply the pigeonhole principle in these cases: The hand has at least four cards in the same suit The hand has at exactly four cards in some ...
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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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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 ...