Question

please do as many as possible

2.25:

Each point in the plane is randomly assigned one of two colors, red

or green. Show that for any real number l > 0, there exist two

points on the plane a distance l apart, that are either both assigned

red or both assigned green. (Hint: consider equilateral triangles.)

2.18:

Suppose you have a circle of radius 1 with center at the origin.

Suppose that 200 points are picked on the circle, of integer degrees,

where by degree of the point, we mean the degree measure of the

angle made by the radius through that point with respect to the

positive x-axis. Show that at least two points must be antipodal,

i.e, at opposite ends of a diameter of the circle. What is the least

number k of points that you can pick as described and still be

guaranteed that you will have a pair of antipodal points?

2.16:

Write down any 10 integers in a list. Label them a1, a2, ... , a10.

Prove that some sequence of consecutive terms ai, ai+1, ai+2, ... ,

aj in your list (with 1  i  j  10) is sure to add to a multiple

of 10. (Note: it is possible that j = i. Hint: let a0 be an arbitrary

integer, and consider the set of integers a0, a0 + a1, a0 + a1 + a2, ... , a0 + a1 + ··· + a10.)

2.14

Given any integer n 1, show that there exists an integer, whose

digits are all either 0 or 1, that is divisible by n. (Hint: you know

that if you take any set of n + 1 integers, the di↵erence of some two

of them must be divisible by n. Now try to choose your set of n + 1

integers carefully!)

Answer 1