# Tagged Questions

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $2^n$ 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 -

Let $n$ be an odd integer and let $f$ be an $n$-permutation of length $n$. Show that the number $$x = (1-f(1))\cdot(2-f(2))\cdot...(n-f(n))$$ is even. I don't understand ...
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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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### 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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### 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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### 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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### 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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### 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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### Prove a number is even using the Pigeonhole Principle

Let n be an odd integer and let f be an [n]-permutation of length n, where [n] is the set of integers 1, 2, 3,...n Show that the number ...
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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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### 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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### 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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### Combinatorics - possibly pigeon hole, 100 by 100 matrix with numbers from 1 to 100

We are given a $100$ by $100$ matrix. Each number from $\{1,2,...,100\}$ appears in the matrix exactly a $100$ times. Show there is a column or a row with at least $10$ different numbers. I'd like a ...
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### Couple of Counting (how many ways) questions.

1.If I have a group of 10 seats reserved for people, and there are n=>10 total people, how many ways are there to choose who gets the 10 seats? for ex:If there was a definite number of people lets ...
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### 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?
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### pigeonhole principle related problem

I'm given the problem: In a tournament which 18 teams participate, a team being matched with another in a round don’t match again in the follwoing (later) rounds. After 8 rounds prove that there are 3 ...
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 ...