# Tagged Questions

Questions involving the pigeonhole principle, which states that if $n$ items are placed in $m$ containers and $n>m$, then one at least one container has more than one item.

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### Pigeonhole principle (I think): colored points in the plane

Suppose that each points in $\Bbb R^2$ is colored red, green or blue. Prove that either there are two points of the same color a distance $1$ unit apart, or there is an equilateral triangle of side ...
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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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### Elegant proof of icosohedron property

This problem was question A1 on the 2013 Putnam contest. Is there a better way to solve this problem than just using pigeonhole principle? Specifically, is there a group theoretic way to interpret ...
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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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### Ramsey Type problem (variant of people at a party)

There is $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\lfloor n/2\rfloor-1$ of them, each of whom either knows both or else ...
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### Select 100 integers from 1,2,…,200 [duplicate]

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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### A problem in discrete math (Pigeonhole principle related)

This question is from the multiple choice test: A circular region is divided by 5 radii into sectors, as shown above. Twenty-one points are chosen in the circular region, none of which is on any of ...
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### Subsets with equal sums

I have a problem to solve but I am in need of your help. Subjects with equal sums: Prove that for every set $A$ which consists of $10$ double digit natural numbers( numbers among $10, \ldots, 99$), ...
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### Pigeonhole Principle - numbers between $1$ and $100$

Of the set $A=${$1,2,...,100$}, we will choose $51$ numbers. Prove that, among the $51$ chosen numbers, there are two such that one is multiple of the other My notes: 1) There are $25$ prime numbers ...
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### Arrangement of $100$ points inside $13\times18$ 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 ...
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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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### Let $S=\{3,4,5,6,7,8,9,10,11,12\}$. Suppose 6 integers are chosen from S. Must there be 2 integers whose sum is 15?

How can I go about solving this Pigeonhole Principle problem? So I think the possible numbers would be: $[3+12], [4+11], [5+10], [6+9], [7+8]$ I am trying to put this in words...
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### Why is the Ramseys theorem a generalization of the Pigeonhole principle

German Wikipedia states that the Ramseys theorem is a generalization of the Pigeonhole principle source But does not say why this is true. I am doing a presentation about the Ramsey theory and also ...
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### How would you prove this theory of computation problem?

I have trouble proving the following statement, I'm supposed to do it for our theory of computation course but since I've been trying for days I'm looking for a hint : What is the smallest value ...
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### For any arrangment of numbers 1 to 10 in a circle, there will always exist a pair of 3 adjacent numbers in the circle that sum up to 17 or more

I set out to solve the following question using the pigeonhole principle Regardless of how one arranges numbers $1$ to $10$ in a circle, there will always exist a pair of three adjacent numbers in ...
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### 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
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### pigeonhole principle problem 3

Prove: For every group of 1009 positive integers, there exist 2 integers of that group, that their sum or difference divide with 2014 without residue. where do I start?
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### Pigeonhole Principle: birthdays on same day of week

How many people must be in a room so that at least 10 have a birthday on a Friday? edit: Assume that no two people share the same birthday I'm somewhat confused and see two different ways to solve ...
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### 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 ...
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### Pigeonhole principle exercises

I have an exam in combinatorics on Friday and the pigeonhole principle is a part of the material. Can someone give me a reference to a book with the hardest(!) questions on this material? Thank you ...
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### Lower bound for the number of coin weightings

The book that I am currently studying has the following exercise. Given is a set of $n$ coins of weights $0$ or $1$ and a scale to weight them. We would like to determine the weight of each coin by ...
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### 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 ...
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### Prove existence of a triangle with least angle $\leq 30$ degrees

Problem: Let $A$ be a set of $6$ points in a plane such that no $3$ are collinear. Show that there exist 3 points in $A$ which form a triangle having an interior angle not $30$ degrees. I am supposed ...
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### Combinatorics problem; counting in two ways, china 1993

I'm trying to solve the combinatorics problems provided in Yufei Zhao's blog. Can you help me with this one? China (1993): A group of $10$ people went to a bookstore. It is known that ...
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### Application of Pigeon-Hole Principle to balls in bins.

Given $n$ balls placed in $m$ boxes, prove that if $n < \frac{m(m-1)}{2}$ then at least two boxes have same number of balls in them.
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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 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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### 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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### 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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### Polygon and Pigeon Hole Principle Question

Seven vertices are chosen in each of two congruent regular 16-gons. Prove that these polygons can be placed one atop another in such a way that at least four chosen vertices of one polygon coincide ...
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### Generalizations of the pigeonhole principle

Let us place the numbers $1,2,3....,10$ in a random order on a circular table with 10 places. The question is: prove that there are three consecutive numbers with a sum of 17 or more. I know that we ...
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### 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 $x \in \mathbb{R}$...
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