Questions involving the pigeonhole principle in Combinatorial Analysis.

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Pigeonhole Principle question - sum of natural numbers

Let $f:\{1,2,...,15\} \rightarrow \Bbb N$ be a function such that $\sum_{i=1}^{15} f(i) =100$. $f(15+1)$ is defined to be $f(1)$. I have shown that $14\leq f(i)+f(i+1)$ for some $1\leq i \leq15$ ...
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Help prove $f:X \rightarrow Y$ is an injection $\Leftrightarrow$ $f:X\rightarrow Y$ is a surjection when $|X|=|Y|$

I need to prove: Given non-empty finite sets $X$ and $Y$ with $|X| = |Y|,$ a function $X\rightarrow Y$ is an injection if and only if it is a surjection. The hint given is to use the pigeonhole ...
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A problem about pigeonhole principle or graph.

Let $A=\{1,2,...,n\}$, where $\binom{n}{3}\geq n+1$. Let $A_1,A_2,...,A_{n+1}$ be distinct subsets of $A$ such that $\bigcup_{i=1}^{n+1}A_i=A$ and $n(A_i)=3$ for all $i$. How to prove or disprove that ...
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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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Pigeonhole Principle and maximum length of the repeating section

The question I have is, when 5 / 20483 is written as a decimal, what is the maximum length of the repeating section of the representation? I believe I need to divide 5 by 20483 which is equal to ...
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Prove that there is an element in $ \{ \sqrt{3}, 2\sqrt{3}, 3\sqrt{3},…\}$ 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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pigeonhole principle on a circle

In a disk of radius 10, how can we find all values n such that there are exactly n points in the disk and such that no matter how the n points are arranged, we can draw a disk with radius 1 in the ...
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Factorial Divides Rising Power Proof Help

I'm trying to prove the following: $m^{\overline n} \equiv 0 \bmod n!$ Where $m^{\overline n} = m\left({m+1}\right)\left({m+2}\right)\ldots\left({m+n-1}\right)$, the product of $n$ successive ...
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Pidgeonhole Principle.

Suppose there are 3000 members in each of the club X, Y and Z. Each member from each of these three clubs has at least 3001 friends from the other two clubs altogether. Show that there are three ...
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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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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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Midpoints joining integers on a plane lattice

How can you prove that if five nodes of a plane lattice are chosen at random then, the midpoint of the segment between the two points is a lattice point.
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Whether it is the pigeonhole principle?

I had a question, and I am just wondering if it is a question that involves combinations/permutations or the pigeonhole principle. In a class are $20$ students. What is the probability that at least ...
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Square labeled with same number.

Recently I met this combinatorics problem: "Let all points with integer coordinates in a plane be labeled with one of the numbers $1,2,3,...,n$. Prove that there is a rectangle whose vertices are ...
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Solution Verification for How Many Class Rooms Are Needed

The Question There are 38 different time periods during which classes at a university can be scheduled. If there are 677 different classes, how many different rooms will be needed? My Work There ...
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Pigeon-hole principle applied to the game of tic tac toe

In a game of tic tac toe, noughts and crosses are drawn inside an unoccupied cell of a 3 x 3 matrix by two players I, II in alternating moves. Player I draws crosses and Player II draws noughts. The ...
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Pigeon Hole Principle (involving distances)

There are 100 old(non-digital) watches in an antique shop, all running but not necessarily on time. Prove that at some moment of time the sum of the distances from the center of the shop to the ...
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Pigeonhole principle questions

I want to solve the following problems with Pigeonhole principle. Show that in every group of people that have atleast 2 people, we can find couple that know the number of the people in the group.( ...
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How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

This a shortened version (motivation from telecommunications stripped away) of a question I asked in MO in late May (no answers). I am mostly checking, if somebody has seen this or a related question ...
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Improving statement obtained by Pigeonhole principle

In this MSE question, this statement is proven: Room is cube-shaped, with side 3m. 136 flies fly in it. Prove that at any moment one can encompass 6 flies with a sphere of radius 90cm. Can ...
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Using the pigeonhole principle to prove there is at least a sum of numbers bigger than 29.

There is a circumference with 14 points $\{p_{1}, p_{2}, ... p_{14}\}$. These points are assigned numbers 1 to 14 randomly. It must be proven that if points are taken three-by-three, these triplets ...
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Seeking Feedback on my solution: Maximum area of a triangle inside a rectangle.

I was solving problem 3.3.16 from Paul Zeitz's book "The Art and Craft of Problem Solving." The problem reads Inside a 1 x 1 square, 101 points are placed.Show that some three of them form a ...
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Need a hint with permutations and pigeonhole-principle question

let $\pi_1,\pi_2,\pi_3\in S_{28}$. Help me prove that there are two sub-sequences of 28 with length 4 $i_1< i_2 <i_3<i_4,\ and\ \ j_1<j_2<j_3<j_4$ so that $\pi_q(i_n)=\pi_p(j_n)$ ...
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2014 points inside a cube

$2014$ points are chosen inside a cube with side $13$. Can a cube with side $1$ be found inside it so that it doesn't contain any of chosen points? This must be a problem solved using ...
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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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Prove that $ax^2 + by^2 \equiv c \ ( \mod{p})$ has integer solutions

Let $p$ be a prime number and $a, b, c$ integers such that $a$ and $b$ are not divisible by $p$. Prove that $ax^2 + by^2 \equiv c \ ( \mod{p})$ has integer solutions Well, this problem can be ...
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Pigeon hole principle:Trominoes and chessboards

Heres the question: What is the largest number of squares on an 8 $\times$8 checkerboard which can be colored green,so that in any one arrangement of three squares ("tromino"),at least one square ...
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$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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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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Friend Group and Hater Group

Consider a set $S$ of $n$ people such that, for all distinct $x$ and $y$ in $S$, it is the case that either $x$ and $y$ like each other or $x$ and $y$ hate each other. Let us call $S' \subseteq S$ a ...
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Pigeon Hole Principle : Can one of the games be played?

Q: 9 people are in a club. Each of them can play one of the games among Bridge , Hearts & Mahajong. Prove that they can play at least one of the mentioned games.( all games require 4 players.) I ...
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Pigeon Hole Principle - proof of d as a positive integer

Let $d$ be a positive integer and consider any set $A$ of $d+1$ positive integers. Show that there exists two different numbers $x, y\ \epsilon\ A$ so that $ x \mod\ d = y \mod\ d$ and $x =/= y$. ...
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combinatorics - pigeonhole principle - 2

I've advanced a little with this question but I'm not sure that I'm in the right direction. For any set $X$ with $n$ positive numbers, $n>5$, prove the existiance of subset $Y \subset X$ so that ...
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How was the expression derived?

Prove that in any set of ten different two-digit numbers one can select two disjoint subsets such that the sum of numbers in each of the subsets is the same? (Reference, CRUX magazine March 1975). I ...
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mantissa of pi and pigeonhole

It might be worth noting that this is a "pigeonhole principle" problem, but I'm not sure how to use PHP with it. The mantissa of pi is the fractional part of it (i.e. everything after the decimal ...
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four point in a row

We have painted all dots of page with two colors(blue and green), proof that there are four point with green color in a line that distance of any two neighbors of this four is one unit or there are ...