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Lies your teachers have told you Vol1: There is no "formula" for prime numbers. A roughly MX2 level question attached. PS, quicker ways to derive such formulae certainly exist. As an easier side exercise which involves no modular arithmetic, try to construct a formulae for the n-th prime...

On grid paper, a lattice point is a point which is at the intersection of two perpendicular lines. Prove by induction or otherwise that a polygon drawn on grid paper with lattice point vertices has area given by: A = i + b/2 -1 where i is the number of lattice points interior to the polygon...

Anyone up for some multiplayer AoE2? I am horrible at the game but its still pretty fun, especially in big games with teams. (I use the original game plus this expansion http://www.forgottenempires.net/ which contains a few more civs and other features and adds HD support, rather than the...

A positive integer is written on a blackboard. Players A and B take turns subtracting a positive square integer from it such that the result is still positive. The person who plays the last legal move wins. There are ~180,000 out of the first 40,000,000 positive integers that result in a win...

Fun little exercise for those with some familiarity with convergence tests for integrals. (A first year university course in calculus should suffice). $For which real $\alpha,\beta\in\mathbb{R}$ with $\beta>0$ does the improper integral: \\ \\ $\int_0^\infty x^\alpha \sin(x^\beta)\, dx$\\...

People seem to like game theory on here. Here is a nice and simple game that shows some of the mathematics present in poker (beyond mere probability calculations). Suppose there are three cards in a deck, A > K > Q. Players 1 and 2 are each dealt a card out of this three card deck. Each player...

Ten of Chad O Dude's top crushes are in a large field armed with one shot pistols. Each pair of girls is separated by a distinct distance. Each girl shoots the closest girl to her in an attempt to eliminate the competition. What is the largest number of women Chad can end up with?

$For an arbitrary sequence: $u=(u_1,u_2,\ldots)$ of complex numbers, define its sequence of differences by: $u'_n:=u_{n+1}-u_n$.\\ Let $u_n^{(k)}$ denote the $n$-th term of the sequence obtained by taking the sequence of differences of the sequence $u$ $k$-times recursively. (So for example...

Stumbled across a pretty funny mathematical drinking game / problem in graph theory posted by someone on Terence Tao's blog: Say you have a group of n people. The player who is "it" says an integer k strictly between 1 and n+1 and at the same time everyone including this player points at...

Prove that for all positive integers n, the polynomial: P_n(x)=\sum_{k=0}^{2n}\frac{x^k}{k!} has no real roots.

Let a,b,c > 0 with abc=1. Prove that: \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+\frac{c}{(c+1)(a+1)}\geq 3/4.

a) Find a condition for the cubic polynomial x^3+Ax+B to have three real roots (where A,B are real). Let R be the region bounded by the parabola x^2=4y and the line y=20. b) Using a) or otherwise, find the probability of being able to draw three distinct normals from a randomly selected point...

Let a,b,n be positive integers. Prove by induction or otherwise that if n divides a+b and n^2 divides a^2 + b^2, then n^m divides a^m + b^m for all positive integers m.

Chad O. Dude is on a date with a girl at an Italian restaurant. They are sharing a bowl of spaghetti which contains exactly n pieces. They eat by simultaneously choosing an end of a piece of spaghetti and sucking it into their mouths, then repeating this process. Find the probability of Chad...

Attached is a handful of somewhat difficult MX2 level questions written by me for a BoS meat a while back. Some of you will have seen them before, but I never wrote up solutions here (and don't really have the time to do so now). If you think you have solved any of them feel free to post here...

Carrot, have you ever seen the really sexy nonconstructive proof that you can find irrationals a,b such that a^b is rational?

$A group of $23$ people are in a room.\\ \\ Prove that it is more likely than not that there is a common birthday within the group. (Ie that two or more of them have the same birthday). You may assume that there are 365 equally likely possibilities for each person's birthday.$

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Anyone here play ssbb? I haven't played it in years but am feeling like messing around online.

Five hundred pirates stumble across a chest filled with 100 pieces of gold. Being in a hierarchical society, the pirates have ranks #1 to #500. The pirate king (#1) decides on a way of distributing the gold. Eg 50 for him, 50 for #2. That night, the pirates all take a vote on whether this...