# 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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### Show that, of any set of $2^{n+1}-1$ positive integers numbers is possible choose $2^{n}$ elements such that their sum is divisible by $2^{n}$.

Show that, of any set of $2^{n+1}-1$ positive integers numbers is possible choose $2^{n}$ elements such that their sum is divisible by $2^{n}$. My approach: Let $a_1,\ldots,a_{2^{n+1}-1}$ positive ...
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### Least distance between two points in an equilateral triangle [closed]

Five points lie inside an equilateral triangle of side 2 units.Prove that at least 2 points are no more than a unit distance apart.
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### Jessica the Combinatorics Student, part 2

The original question about Jessica, which I encourage review of, is as follows: Jessica is studying combinatorics during a $7$-week period. She will study a positive integer number of hours every ...
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### Pigeonhole Principle Question: Jessica the Combinatorics Student

Jessica is studying combinatorics during a $7$-week period. She will study a positive integer number of hours every day during the $7$ weeks (so, for example, she won't study for $0$ or $1.5$ ...
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### Prove there's a monochromatic isosceles triangle.

The points in a circle are coloured red and blue. Prove that there exists a monochromatic isoceles triangle. I can prove that there exists a monochromatic triangle. If there are no three points of ...
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### Chessboard Kings and no check [closed]

What is the largest number of kings which can be placed on a chessboard so that no two of them put each other in check?
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### Application of Jacobi's Theorem in Box Principle

Today I was going through Problem Solving Strategies by Arthur Engel, and found this in the chapter Box Principle Before the question it says it "treats a theorem of Jacobi and its applications" ...
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### Proof related to pigeon hole principle to be done with induction

since the question is about a positive integer m, it's obvious that the use of mathematical induction needed, but to prove the fact for n = k+1 we have to use the pigeon hole principle, i am so ...
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### how to find pigeon holes in a question related to pigeon hole principle

prove that every set of 10 two digit numbers has two disjoint subsets with the same sum of elements. In this question i don't know how to choose the pigeon holes, or what will be the pigeon holes
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### Distance between points in a square [closed]

Let $p$ be a square whose side has length $1$. $51$ points are randomly chosen inside the square. Show that there are atleast $3$ points whose mutual distance is $< \sqrt{0.08}$.
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### Proving the Pigeonhole Principle

I am looking to prove the Pigeonhole Principle by proving the following claim: Let $A$ be a set with $m$ elements, and let $B$ be a set with $n$ elements, where $m,n\in \omega$ and $m > n$. ...
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### A group of $15$ boys plucked a total of $100$ apples. Prove that two of those boys plucked the same number of apples.

A group of $15$ boys plucked a total of $100$ apples. Prove that two of those boys plucked the same number of apples. My answer is: First distribute $90$ apples so that each will have $6$ apples. ...
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### Russia (2000) contest:Prove the existence of a pair of rows and columns with intersections differently coloured

We have a $100\times100$ board divided into $10^4$ unit squares. These squares are coloured with four colours so that every row and every column has $25$ squares of each colour. Prove that there are ...
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### Show that a class of nine students must have at least three male students or at least seven female students. [closed]

I am stuck with the following problem: Suppose that there are nine students in a class. Show that the class must have at least three male students or at least seven female students. Please help me ...
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### Applying pigeonhole principle to determine whether a list of strings must have duplicates.

Say you have a program that creates strings of lower-case letters of length 5 or less. It is told that the program holds 600,000 words on its drive. How can you figure out if all the words are ...