please do as many as possible

2.25:

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

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

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

red or both assigned green. (Hint: consider equilateraltriangles.)

2.18:

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

Suppose that 200 points are picked on the circle, of integerdegrees,

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

angle made by the radius through that point with respect tothe

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

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

number k of points that you can pick as described and stillbe

guaranteed that you will have a pair of antipodalpoints?

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 amultiple

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

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 ofsome two

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

integers carefully!)