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High School
By
True IMO
on Nov. 14, 2016 | Updated Jan. 4, 2018
International Mathematical Olympiad 2016 Problem 5
The equation \((x-1)(x-2)(x-3)...(x-2016) = (x-1)(x-2)(x-3)...(x-2016)\) is written on a board, with 2016 linear factors on each side. What is the least possible value of \(k\) for which it is p…
Mathematics
Combinatorics
imo
imo 2016
polynomials
0
Undergraduate
By
Shiva Kintali
on May 31, 2012 | Updated Dec. 6, 2017
Party Problem
Suppose there are six people at a party. Prove that there are always three of them so that every two know each other (or) no two know each other. In other words, let the edges of the complete graph o…
Mathematics
Puzzles
Combinatorics
Graph Theory
Puzzles
counting
extremal graph theory
interview question
0
Undergraduate
By
Shiva Kintali
on Aug. 29, 2013 | Updated Jan. 4, 2018
Even Subsets
A set \(T\) is called even if it has even number of elements. Let \(n\) be a positive even integer, and let \(S_1, S_2, \dots, S_n\) be even subsets of the set \(S = \){\(1,2,\dots,n\)}. Prove th…
Mathematics
Combinatorics
counting
0
High School
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Basics of counting
Let \(S = {1,2,...,n}\). How many ordered pairs \((A,B)\) of subsets of \(S\) are there that satisfy \(A \subseteq B\) ?
Mathematics
Combinatorics
basics
counting
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 June 6, 2012 | Updated Dec. 6, 2017
Happy Ending Problem
Prove the following : Any set of five points in the plane in general position has a subset of four points that from the vertices of a convex quadrilateral. For any positive integer \(N\), any suffic…
Mathematics
Combinatorics
Geometry
counting
0
High School
By
Shiva Kintali
on Sept. 27, 2013 | Updated Jan. 4, 2018
Binomial coefficients
Evaluate the following sums using combinatorial methods and algebraic methods : \(\displaystyle \sum_{i=0}^{k} {m \choose i}{n \choose k-i}\) \(\displaystyle \sum_{i=0}^{n} {n \choose i}^2\) …
Mathematics
Combinatorics
binomial theorem
counting
n choose k
0
High School
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Bijective counting
Let \(S = {1,2,...,n}\). How many ordered pairs \((A,B)\) of subsets of \(S\) are there that satisfy \(A \subseteq B\) ? Let \(S = {1,2,...,n}\). How many ordered pairs \((A,B)\) of subsets of …
Mathematics
Combinatorics
bijection
counting
0
Undergraduate
By
Shiva Kintali
on June 6, 2012 | Updated Dec. 6, 2017
Distances between points in a plane
Let \(S\) be a set of \(n \geq 3\) points in the plane such that the distance between any two points is at least 1. Prove that there are at most \(3n - 6\) pairs of points at distance exactly 1. Pro…
Mathematics
Combinatorics
Geometry
counting
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
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