BMAD

Alice and Bob are going to play a game, with the following rules: 1st Alice picks a probability p, 0 <= p < 0.5 2nd Bob takes any finite number of counters B. 3rd Alice takes any finite number of counters A. These happen in sequence, so Bob chooses B knowing p, and Alice chooses A knowing p and B. A series of rounds are then played. Each round, either Bob gives Alice a counter (probability p) or Alice gives Bob a counter (probability 1-p). The game terminates when one player is out of counters, and that player is the loser. Whom does this game favor?

on the right track

good point!

True but this task is about moving as slow as possible. Though he can travel that fast, which is how he is able to deliver all those christmas gifts, we want to know what is the slowest he could go.

@bonanova The path/strategy that allows Santa to continiusly travel around while remaining forever at "night" going at the minimum speed possible.

Is this the slowest?

i changed the status of this question as we consider Rainman's argument.

To be clear.... the best path is the one that provides Santa the slowest speed (gives more time for gift giving)

Assume that the earth is a perfect sphere with a circumference of 40Mm. Santa needs to travel from the North Pole to the South Pole while avoiding daylight. Assuming that he can go faster than the speed of light, what path would be the best path to take and what is the slowest he can travel?

There's a gun located on an infinite line. It starts shooting bullets along that line at the rate of one bullet per second. Each bullet has a velocity between 0 and 1 m/s randomly chosen from a uniform distribution. If two bullets collide, they explode and disappear. What is the probability that at least one of the bullets will infinitely fly without colliding with another bullet?

yes.

I have a smaller shape but this is a good start.

Could you form a generalization as to why? the title is a hint at this,

In an presidential election between Bipa and Viktor, the winning candidate Bipa received n+k votes, whereas Viktor has received n votes. (n and k are positive integers.) If ballots are counted in a random order, what is the probability that Bipa's accumulating count will always lead his opponent's, and why?

Alright, I'll add a puzzle to the bunch... You have N computers on a space station. An accident happens, and some of the computers are damaged, but you know the number of good (undamaged) computers is greater than the number of bad (damaged) ones. Your goal is to find *one* computer that's still good. Your only method of testing is the following: Use one computer (say, X) to test another (Y). If X is a good computer, it tells you correctly the status of Y. If X is bad, it may or may not give the correct status of Y; assume it will give whatever answer is least useful to your testing strategy. In worst-case, how many tests must you use to find one computer that's still good? (in terms of N) You're permitted any combination of tests, though keep in mind the bad machines may not be consistent in the results they give you.

There are three one-dimensional tracks, of length 12, 7, and 5 spaces respectively. You start with pennies in the first space of each track; your opponent starts with pennies in the last space of each track. On your turn, you may move any one of your pennies any number of spaces in either direction along a track (as a chess rook), however you are not permitted to bypass the other player's penny or occupy its space. If a player has no legal move, he loses. What should your first move be?

There are two fractions, 34/55 and 55/89. We are looking for a third fraction of positive integers a/b, where 34/55>a/b>55/89 and 55<b<89. What is the smallest b where this is possible?

Define semiregular polygon as a polygon which has all of its' edges of the same length. Also, all of its' interior or exterior angles must be equal (meaning that any interior angle must be x or 360-x). It must be concave and simple (it should not self-intersect) and only two of its' edges are allowed to meet in each corner. Find the semiregular polygon that has the minimum number of edges.

Get two random numbers between 0 and 1. Subtract the smaller from the bigger. What is the probability that the result is <0.3? Is it bigger than the probability it is > or equal to 0.3? For what number the probability for both cases is the same?

Given a triangle whose three sides are consecutive integer values, and the area of which is divisible by 20, find the smallest possible side for which these conditions hold true: two sides are odd numbers at least one side is a prime number. The added condition to my original gets into some cool number theory (if you go that route)

Trying to help with the difficulties of language ... Should we find a triangle, that meets the conditions, that has a side that is smaller then the smallest side of any other triangle that meets the conditions? yes, when you phrase it that way, i see how complex it sounds. Ahh the joys of the English language. Seems much more clear in my native language

Phil I don't belive your answers meet all the conditions

very well done Rainman. For the proof to the last part that you mentioned,