# Search results

1. ### Geography of the shahnameh

I'm reading a translation of the shahnameh, which says that Afrasiab was a Turanian leader, who upon failed invasions of Iran, retreats beyond the Jihun river. But wikipedia says https://en.m.wikipedia.org/wiki/Ceyhan_River The Jihun river is a river in turkey. But Turan...
2. ### I Proving the product rule using probability

I thought this was kind of a cool proof of the product rule. Let ##F(x)## and ##G(x)## be cumulative distribution functions for independent random variables ##A## and ##B## respectively with probability density functions ##f(x)=F'(x)##, ##g(x)=G'(x)##. Consider the random variable...
3. ### Challenge 14: Smooth is not enough

A function f:\mathbb{R} \to \mathbb{R} is called "smooth" if its k-th derivative exists for all k. A function is called analytic at a if its Taylor series \sum_{n\geq 0} \frac{f^{(n)}(a)}{n!} (x-a)^n converges and is equal to f(x) in a small neighborhood around a. The challenge...
4. ### Confidence Interval for Coefficient of Quadratic Fit

I have a bunch of noisy data points (x,y), and I want to model the data as y = ax2 + bx + c + noise where noise can probably be assumed to be Gaussian, or perhaps uniformly distributed. My data is firmly inside of an interval and I'm only interested in modeling correctly inside of this...
5. ### Challenge 13: Sums of Sines

Prove that \sum_{k=0}^{n} \sin\left( \frac{k \pi}{n} \right) = \cot \left( \frac{\pi}{2n} \right)
6. ### Challenge 12b: Flipping Coins

A man has a single coin in front of him that lands on heads with probability p and tails with probability 1-p. He begins flipping coins with his single coin. If a coin lands on heads it is removed from the game and n new coins are placed on his stack of coins, if it lands on tails it is...
7. ### Challenge 12a: Flipping Coins

A man begins flipping coins according to the following rules: He starts with a single coin that he flips. Every time he flips a heads, he removes the coin from the game and then puts out two more coins to add to his stack of coins to flip, every time he flips a tails he simply removes the coin...
8. ### Challenge 11: Sequence of Primes

Consider the following sequence: a1 = p, where p is a prime number. an+1 = 2an+1 Prove there is no value of p for which every an is a prime number, or make me look dumb and construct a counterexample.
9. ### Challenge 10b: Temperatures of the Earth

For the sake of this challenge, you can assume that the world behaves nicely (everything is continuous, differentiable etc, the world is a sphere blah blah blah) but figurative bonus points if you assume fewer things. If you're looking for an easier challenge check out Challenge 10a. The...
10. ### Challenge 10a: Temperatures of the Earth

For the sake of this challenge, you can assume that the world behaves nicely (everything is continuous, differentiable etc, the world is a sphere blah blah blah) but figurative bonus points if you assume fewer things. If you're looking for a bigger challenge check out Challenge 10b. The...
11. ### Challenge 9: The Prisoner Line-Up

Countably infinite prisoners, numbered 1,2,3 etc. are told that tomorrow they will be given hats, colored either black or white, and lined up so that each prisoner can only see the prisoners whose hats are labeled with a greater number. The prisoners then get to guess which color hat they wear...
12. ### Challenge 8: Sub-Gaussian Variables

A random variable X is called sub-gaussian if P( |X| > t) \leq 2e^{-K t^2} for some constant K. Equivalently, the probability density p(x) is sub-gaussian if \int_{-t}^{t} p(x) dx \geq 1 - 2 e^{-Kt^2}. The challenge: Prove that the standard normal distribution (with mean 0 and...
13. ### Challenge 7b: A Snail's Pace

This question is courtesy of mfb: An immortal snail is at one end of a perfect rubber band with a length of 1km. Every day, it travels 10cm in a random direction, forwards or backwards on the rubber band. Every night, the rubber band gets stretched uniformly by 1km. As an example, during the...
14. ### Challenge 7a: A Snail's Pace

Similar to the previous two-part question, if you find part b to be an appropriate challenge please leave part a to those who are appropriately challenged by it. This question is courtesy of mfb An immortal snail is at one end of a perfect rubber band with a length of 1km. Every day, it...
15. ### Challenge 6: Gambler's Hell

Sorry about missing a week guys. A degenerate gambler dies and is sentenced to hell. The devil informs him that his punishment is that he must play a slot machine. He starts with one coin, and each time he puts in a coin he gets countably infinite many coins out of the machine. He must...
16. ### Challenge 5: Sum Free Subsets

A set A of non-zero integers is called sum-free if for all choices of a,b\in A, a+b is not contained in A. The Challenge: Find a constant c > 0 such that for every finite set of integers B not containing 0, there is a subset A of B such that A is sum-free and |A| ≥ c|B|, where |A| means the...
17. ### Build up of Ingredients

My girlfriend found several little black cylinders that looked like rodent feces in her Fruit Loops the other day, so she sent them a complaint about it. They replied back saying that the cylinders were "a build up of ingredients" and a couple of coupons that nobody who got mouse poop in their...
18. ### Challenge 4: There's no app for that: integration

a.) Poor Wolfram Alpha got asked to calculate the following integral \int_{0}^{\infty} e^{-ax} \frac{\sin(x)}{x} dx but couldn't handle it! http://www.wolframalpha.com/input/?i=int_%7B0%7D%5E%7Binfty%7D+e%5E%7B-ax%7D+sin%28x%29+%2Fxdx (Results are not guaranteed if you use wolfram alpha...
19. ### Weekly Math Challenge Point Standings

Here are the total point standings as of 10:33am on 12/20/2013: mfb 9 Citan Uzuki 7 Boorglar 3 jbunniii 3 Mandelbroth 2 jk22 2 hilbert2 2 economicsnerd 2 HS-Scientist 1 D H 1 jackmell 1 verty 1 Perok 1
20. ### Challenge 3b: What's in a polynomial?

In order to challenge a broader section of the forum, there is a part a and a part b to this challenge - if you feel that part b is an appropriate challenge, then I request you do not post a solution to part a as part a is a strictly easier question than part b. The new challenge: Are there any...
21. ### Challenge 3a: What's in a polynomial?

In order to challenge a broader section of the forum, there is a part a and a part b to this challenge - if you feel that part b is an appropriate challenge, then I request you do not post a solution to part a as part a is a strictly easier question than part b. The challenge: Prove that the...
22. ### Ambiguous email, help!

An email conversation has gone like this: A: Can you come by sometime next week? B: I'm free Thursday or Friday C: "Can you stop by in the morning?" OK, does that mean he wants me to come by next Thursday or Friday in the morning, or is he separately asking me to stop by tomorrow?
23. ### Challenge 2: Covering the Triangle

Suppose A1, A2 and A3 are closed convex sets, and let Δ be a triangle with edges F1, F2 and F3 such that A_1 \cup A_2 \cup A_3 = \Delta and F_i \cap A_i = \emptyset \text{ for } i=1,2,3 Prove there exists some point x\in \Delta such that x\in A_1 \cap A_2 \cap A_3 Figurative bonus...
24. ### Challenge 1: Multiple Zeta Values

A multiple zeta value is defined as \zeta(s_1,...,s_k) = \sum_{n_1 > n_2 ... > n_k > 0} \frac{1}{n_1^{s_1} n_2^{s_2}...n_k^{s_k}} . For example, \zeta(4) = \sum_{n = 1}^{\infty} \frac{1}{n^4} and \zeta(2,2) = \sum_{m =1}^{\infty} \sum_{n = 1}^{m-1} \frac{1}{ m^2 n^2} . Prove the...
25. ### Weekly Math Challenge FAQ

What is this place? This forum is for people to come together and stretch their brains on math puzzles. Each week there will be a new challenge for the forum to try. What do I get for answering challenges? We're going to try a point system. The first person to post a solution will be awarded...
26. ### Blanking on word for kind of convergence of a sum

I have a sum \sum_{n=-\infty}^{\infty} f(n) which I do not want to consider the convergence of in the normal sense, but I want to talk about the limit \lim_{N\to \infty} \sum_{n=-N}^{N} f(n). I know that when this limit exists the sum is _____ convergent, or is a _____ sum, where _____ is...
27. ### Rational Numbers That Are Hard To Prove?

There are lots of examples of numbers where "is it a rational number" has been an open question for a while before being proved as not being rational. Pi and e being famous examples. Some of them are still open, like pi+e, or the Euler-Mascharoni constant, but I think the general consensus is...
28. ### Almost zeta(2)

I'm noticing wolfram alpha has the amazing ability to analytically solve \sum_{n=1}^{\infty} \frac{1}{n^2 + a^2} Anyone know how to do this, and if it's also possible to deal with higher order guys (like it also can do 1/(n4+a2), but it's a way more complicated expression to the point...
29. ### Bilateral trade between all countries

I'm interested in finding a database that contains the bilateral trade (imports/exports) between every pair of countries. I've done some googling and I can't find it all contained in one place. Preferably if it was in a downloadable csv file (which would basically be a godsend for me) but...
30. ### Bravecar is pretty awesome

Haha this is pretty awesome http://www.goblinscomic.com/wp-content/uploads/2012/05/Bravecar.jpg