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Undergraduate
By
123forman
on May 8, 2014 | Updated Dec. 6, 2017
Democratic distribution of wealth
A crew of 100 pirates have captured 100 laptops. The pirates are ranked 1 through 100, with pirate 100 being the pirate king. Captured booty is distributed as follows. The pirate king proposes a dis…
Mathematics
Discrete Mathematics
Game Theory
induction
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
High School
By
Shiva Kintali
on June 27, 2013 | Updated Jan. 4, 2018
Basics of Induction
Prove the following using induction: \(\sum_{i=1}^{n}{i} = \frac{n(n+1)}{2}\). \(\sum_{i=1}^{n}{i}^2 = \frac{n(n+1)(2n+1}{6}\). \(\sum_{i=1}^{n}{i}^3 = {(\frac{n(n+1)}{2})}^2\). …
Mathematics
Discrete Mathematics
induction
summation
0
Undergraduate
By
Vibhav Pant
on March 4, 2014 | Updated Dec. 6, 2017
Prove $\sum_{z=0}^{\infty}(\zeta(z)-1)=1$
Riemann's zeta function \(\zeta(z)\) is defined as \(\zeta(z)=1+\frac{1}{2^z}+\frac{1}{3^z}+\cdots=\sum_{k=0}^{\infty}\frac{1}{k^z}\) Prove that \(\sum_{z=0}^{\infty}(\zeta(z)-1)=1\)
Mathematics
Discrete Mathematics
sums
1
Undergraduate
By
Shiva Kintali
on Sept. 27, 2013 | Updated Jan. 4, 2018
Fibonacci numbers and Induction
The Fibonacci numbers, \(F_0, F_1, F_2, \dots\) , are defined recursively by the equations \(F_0 = 0\), \(F_1 = 1\), and \(F_n = F_{n-1} + F_{n-2},\) for \(n > 1\). Prove that …
Mathematics
Discrete Mathematics
fibonacci
induction
tiling
0
Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Asymptotic Summations
Prove the following : \(\sum_{i=1}^{n} \frac{\log i}{i} = \Theta ((\log n)^2)\) \( \sum_{i=0}^{n} |\sin(i)| = \Theta(n)\) where \(i\) is in radians.
Mathematics
Discrete Mathematics
asymptotic analysis
0
High School
By
Shiva Kintali
on Sept. 30, 2013 | Updated Jan. 4, 2018
Using Binomial Theorem
Prove the following using binomial theorem and/or mathematical induction Let \(a\), \(b\) and \(n\) be natural numbers, prove that \(\frac{(a+\sqrt{b})^n + (a-\sqrt{b})^n}{2}\) is also a natural …
Mathematics
Discrete Mathematics
binomial theorem
induction
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