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High School
By
Shiva Kintali
on Aug. 28, 2013 | Updated Jan. 4, 2018
7 points inside a hexagon
Consider a hexagon \(H\) with side length 1. Given any 7 points inside \(H\), show that at least two points are separated by a distance of at most 1.
Puzzles
Puzzles
geometry puzzle
pigeonhole principle
0
High School
By
Shiva Kintali
on May 7, 2014 | Updated Jan. 4, 2018
Sock Drawer puzzle
There are 10 socks of each of the following colors in a drawer: red blue green black white i.e., there are 50 socks. The socks are arbitrarily distributed in the drawer. You are blind-f…
Puzzles
Puzzles
math puzzle
pigeonhole principle
0
Undergraduate
By
Shiva Kintali
on June 13, 2012 | Updated Dec. 6, 2017
Basics of Pigeonhole Principle
Prove the following : Given \(n\) integers, some nonempty subset of them has sum divisible by \(n\). Let \(A\) be a set of \(n+1\) integers from {\({1, 2,\dots , 2n}\)}. Prove that some element of \…
Mathematics
Discrete Mathematics
basics
counting
pigeonhole principle
1
Undergraduate
By
Shiva Kintali
on Sept. 30, 2013 | Updated Jan. 4, 2018
Friends and Parties
Show that at a party of \(n\) people, there are two people who have the same number of friends in the party. Assume that friendship is symmetric. There are \(2n\) people at a party. Each person has…
Mathematics
Discrete Mathematics
counting
pigeonhole principle
0
Undergraduate
By
Shiva Kintali
on Sept. 29, 2013 | Updated Jan. 4, 2018
Fibonacci numbers and primes
The Fibonacci numbers are defined by \(F_1 = F_2 = 1\) and \(F_n = F_{n−1} + F_{n−2}\) for \(n \geq 3\). If \(p\) is a prime number, prove that at least one of the first \(p + 1\) Fibonacci numbers mu…
Mathematics
Combinatorics
fibonacci
pigeonhole principle
primes
0
Undergraduate
By
Shiva Kintali
on Sept. 29, 2013 | Updated Jan. 4, 2018
Dirichlet approximation
(Trivial approximation) For \(x \in \mathbb{R}\) and \(n \in \mathbb{Z}^+\), there is a rational number \(\frac{p}{q}\), with \(1 \leq q \leq n\), such that …
Mathematics
Combinatorics
pigeonhole principle
0
Graduate
By
Shiva Kintali
on June 6, 2012 | Updated Dec. 6, 2017
Erdos-Szekeres Theorem
Prove that for given integers \(r, s\) any sequence of distinct real numbers of length at least \((r - 1)(s - 1) + 1\) contains either a monotonically increasing subsequence of length \(r\), or a mon…
Mathematics
Combinatorics
pigeonhole principle
0
Undergraduate
By
Shiva Kintali
on June 13, 2012 | Updated Dec. 6, 2017
Ramsey Number upper bound
The Ramsey Number \(R(s, t)\) is the minimum integer \(n\) for which every red-blue coloring of the edges of a complete graph \(K_n\) contains a completely red \(K_s\) or a completely blue \(K_t\). Pr…
Mathematics
Graph Theory
pigeonhole principle
0
Undergraduate
By
Shiva Kintali
on July 26, 2013 | Updated Jan. 4, 2018
Equal degree vertices
Prove that every simple graph with at least two vertices has two vertices of equal degree. Is the conclusion true if we allow multi-edges ?
Mathematics
Graph Theory
pigeonhole principle
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