For the FIFA world cup, Paul the octopus has been predicting the winner of each match with amazing success. It is rumored that in a match between 2 teams A and B, Paul picks A with the same probability as A's chances of winning.Let's assume such rumors to be true and that in a match between X and Y, X the stronger team has a probability of 2/3 of winning the game. What is the probability that Paul...

Given a collection of points P in the plane, a 1-set is a point in P that can be separated from the rest by a line; i.e. the point lies on one side of the line while the others lie on the other side. The number of 1-sets of P is denoted by n1(P). Find the maximum value of n1(P) over all configurations P of 10 points in the plane.

A sheet of paper has statements numbered from 1 to 35. For all values of n from 1 to 35, statement n says "At most n of the statements on this sheet are false". Which statements are true and which are false?

A and B play the following min-max game. Given the expressionN = 12 + X*(Y - Z)where X, Y and Z are variables representing single digits (0 to 9), "A" would like to maximize N while "B" would like to minimize it. Towards this end, "A" chooses a single digit number and "B" substitutes this for a variable of her choice (X, Y or Z). "A" then chooses the next value and "B", the variable to substitute the...

A file is transferred from one location to another in 'buckets'. The size of the bucket is 10 kilobytes. Each bucket gets filled at the rate of 0.0001 kilobytes per millisecond. The transmission time from sender to receiver is 10 milliseconds per bucket. After the receipt of the bucket the receiver sends an acknowledgement that reaches sender in 100 milliseconds. Assuming no error during transmission,...