Questions involving the pigeonhole principle in Combinatorial Analysis.

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Putnam PigeonHole

This is from page 12 of Putnam and Beyond. Problem: Prove that for every set $X ={x_1,x_2, \ldots ,x_n}$ of $n$ real numbers, there exists a nonempty subset $S$ of $X$ and an integer $m$ such that ...
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Prove by using Pigeon Hole Principle

Let $k \in \mathbb Z^+ $. Prove that there exists a positive integer $n $ such that $k|n$ and the only digits in $n$ are 0's and 3's
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2answers
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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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1answer
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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!
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1answer
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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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2answers
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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
40 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
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Choosing $2n-1$ points from $n\times n$ grid such that $3$ points always form a right triangle

NOTE: Looking for a hint,not the whole solution. BdMO 2012 Nationals Secondary Consider a $n×n$ grid of points. Prove that no matter how we choose $2n-1$ points from these, there will always ...
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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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0answers
48 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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3answers
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Pigeon Hole Principle Algorithm

The “pigeonhole principle” states that if n+1 objects (e.g., pigeons) are to be distributed into n holes then some hole must contain at least two objects. This observation is obvious but useful. ...
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20answers
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What is your favorite application of the Pigeonhole Principle?

The pigeonhole principle states that if $n$ items are put into $m$ "pigeonholes" with $n > m$, then at least one pigeonhole must contain more than one item. I'd like to see your favorite ...
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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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Pigeon hole principle question of dates and days and year.

In last decade in one febuary,5 children were born on same day of the week,but on different dates.explain how can this happen?what are the different date on they are born?If the new year day that year ...
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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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1answer
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Set theory: Metamath Proof of the Pigeon-Hole Principle, Error?

I have recently came discovered Metamath. Supposedly the language is one that a computer may proof-check. I then began to look at concepts that I am familiar with, and decided to look up the pigeon ...
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6answers
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Of any 52 integers, two can be found whose difference of squares is divisible by 100

Prove that of any 52 integers, two can always be found such that the difference of their squares is divisible by 100. I was thinking about using recurrence, but it seems like pigeonhole may also ...
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2answers
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combinatorics - permutations question, possibly with pigeon hole

Let $A \in Mat_n(\mathbb R)$ such that $\forall i,j: a_{ij}\geq 0$ We are given: $$\forall j: \sum_{i=1}^n a_{ij}=\sum_{i=1}^n a_{ji}=1$$ show there's a permutation $\pi \in S_n$ such that $$\forall ...
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1answer
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If one eats $100$ chocolates in $58$ days,then he must be eating exactly 15 chocolates in some consecutive days

BdMO 2014 Nationals $X$ eats 100 chocolates in 58 days,eating at least 1 chocolate per day.Prove that,in some consecutive days,she ate exactly 15 chocolates. I tried using the pigeonhole ...
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2answers
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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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1answer
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$[x-\frac{1}{n}, (n-1)x+\frac{1}{n}]$ 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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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
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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
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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 ...
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1answer
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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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1answer
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Issue concerning enumerating vertices in a prism (number of two adjacent vertices can only differ by a certain amount)

There are 100 vertices in a prism with a 50-gon as its base. Those vertices are assigned integers 1 to 100 (inclusive) in a random order. Each number can only be assigned once. The objective is to ...
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Is there a Pigeon hole principle proof

Let $a_i$, $1 \leq i \leq 5$ denote five positive real numbers such that $\sum_{i =1}^{5}a_i = 100$. Show that there exist a pair $a_i,a_j$ such that $|a_i-a_j|\leq 10$. Is there a proof using pigeon ...
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Pigeonhole Principle and Equivalence Classes

Let $A$ be a finite set with $n \geq 4$ elements and let $\rho$ be an equivalence relation on $A$. Suppose that there are exactly four equivalence classes: $C_1, C_2, C_3, C_4$. Moreover we know that ...
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3answers
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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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4answers
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Pigeon Hole. 80 numbered balls

We have $80$ numbered balls(From $1$ to 80).Among which are $45$ blue and $35$ orange. Prove that at least two blue balls differ by $9$. For example $13$ and $22$ or $69$ and $78$. So they can differ ...
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2answers
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Pigeon hole principle with sum of 5 integers

Prove that from 17 different integers you can always choose 5 so the sum will be divisible by 5. I tried with positive,negative numbers. Even, odd numbers etc but can't find the solution. Any ...
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1answer
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Math and chess question!

Given a $6\times6$ chess board with $13$ marked squares, can you always place three mutually non-attacking rooks on the marked squares? If so, how can this be proven?
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1answer
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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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Prove that the product of primes in some subset of $n+1$ integers is a perfect square.

I am trying to prove the following: The set $A$ consists of $n + 1$ positive integers, none of which have a prime divisor that is larger than the $n$th smallest prime number. Prove that there ...
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0answers
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How to estimate pigeonhole principle?

I was thinking about this after my professor mentioned the pigeonhole principle in class. Let's say we have $N$ items and $M$ containers. Here we assume $N > M$. We will randomly place each of the ...
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3answers
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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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1answer
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Pigeon hole principle?

A guy reads a book. He read for 81 hours last 10 days. Prove that there has been two consecutive days when he read for at least 17 hours. 81 hours / 10 days equals 8,1 hour a day. 2 * 8,1 = 16,2. It ...
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1answer
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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 ...
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2answers
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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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pigeonhole principle For 1-1 and onto function

$|A|=|B|=n\in N$ Prove/Disprove that if $f:A \rightarrow B$ is 1-1 then $f:A \rightarrow B$ is onto. The answer I saw used the Pigeonhole Principle, I am trying to prove it without the principle. ...
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Pigeonhole Principle -

Let $n$ be an odd integer and let $f$ be an $n$-permutation of length $n$. Show that the number \begin{equation}x = (1-f(1))\cdot(2-f(2))\cdot...(n-f(n))\end{equation} is even. I don't understand ...
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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: ...
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1answer
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Proving something using Pigeonhole Principle [duplicate]

How do I prove the following using the Pigeonhole principle? Let $n$ be an odd integer. Prove that there exists a positive integer $k$ such that $2^k \mod n = 1$. I don't understand how I can prove ...
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1answer
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if there are 5 points on a sphere then 4 of them belong to a half-sphere.

If there are 5 points on the surface of a sphere then there is a closed half sphere containing at least 4 of them. It's in a pidgeonhole list of problems, but I think I have to use rotations in more ...
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1answer
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If $n\nmid a,a+d,a+2d. . . a+(n-1)d$,then $(n,d)=1$

None of the numbers in the sequence $a,a+d,a+2d,a+3d. . . a+(n-1)d$ are divisible by $n$.Then we have to prove that d and n are coprime. I am supposed to use the pigeonhole principle for this ...
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1answer
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Combinatorial Question using ramsey's theory or pigeonhole principle??

We are currently going over pigeonhole principle, ramsey's theorem (graphs and such). Stuck on this particular question: Within a group of an odd number of people, show that at least one person knows ...
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3answers
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What Does “less than or equal to 1 apart” Mean?

I thought this question was classified as a word-meaning question. So, does "1 apart" mean 1/2 the side of the triangle? reference:
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1answer
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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
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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 ...