1: Show that in any finite gathering of people, there are at least two people who know the same number of people at the gathering (assume that "knowing" is a mutual relationship).
2: (a) Consider 7 points on the interval [0, 1). Prove that some three of them must all lie within 1/3 of each other.
(b) Inside a unit square, 101 points are placed. Show that some three of them form a triangle with area no more than .01.
3: Show that there is some positive integer of the form 99 · · · 99000 · · · 000 that is divisible by 83.
4: (a) Every point in R3 is colored red, green, or blue. Prove that no matter how the coloring is done, there is a pair of points exactly 1 unit apart that are the same color.
(b) If every point of the plane is colored one of three colors, do there necessarily exist two points of the same color exactly one unit apart?
5: Let B be a set of more than 2n+1/n distinct points with coordinates of the form (±1, ±1, ..., ±1) in n-dimensional space, with n ≥ 3. Show that there are three distinct points in B which are the vertices of an equilateral triangle.
6: Four Points, Two Distances. How many ways can four distinct points be arranged in the plane so that the six distances between pairs of points take on only two different values?