Show a pair has a midpoint with integer coordinates

Let p = {(x1, y1), (x2, y2), (x3, y3), (x4, y4), (x5, y5)} be a set of five distinct points in the plane , each of which has integer coordinates. Show that some pair has a midpoint that has integer coordinates.

Probability: Birthdays on the Same Day

Determine the number of people needed to ensure that the probability at least two of them have the same day of the year as their birthday is at least 70 percent, at least 80 percent, at 90 percent, 95 percent, at least 98 percent, and at least 99 percent.

Application of Set Theoretic Model of Sequences

1. Using the set-theoretic model of sequences, define the following operators, giving their syntax and their semantics: (a) overwrite: given any sequence, s, over a set X, any element, e, of X and any non-zero natural number, n, return a sequence identical to s except that the element at position n is e. For example, overwrite [a, b, c, d] f 3 = [a, b, f, d] (b) insert: : given any sequence, s, over a set X, any element, e, of X and any non-zero natural number, n, return a sequence identical to s except that e is at position n and every element of s at position greater than n has been shifted right by one position. For example, insert [a, b, c, d] f 3 = [a, b, f, c, d] (c) t... click for more

Sets: Element of Q

Show that if a,b is an element of Q then a+b is an element of Q and ab is an element of Q.

Proof that 3^(1/2) is not rational

Prove that 3^(1/2) is not rational.