Finalized Schedule

So I have obtained a copy of the book  Putnam and Beyond by Razvan Gelca and Titu Andreescu. It is a book that teaches a large amount of undergraduate mathematics, primarily for the purpose of preparing for competitions such as the Putnam competition. My tentative plan is to work through this book during the week, and take "tests" on weekends. (A test would be a complete competition of some kind).

The book gives the advice:


If you are a Putnam competitor, then as you go on with the study of the book try
your hand at the true Putnam problems (which have been published in three excellent
volumes). Identify your weaknesses and insist on those chapters of Putnam and Beyond.
Every once in a while, for a problem that you solved, write down the solution in detail,
then compare it to the one given at the end of the book. It is very important that your
solutions be correct, structured, convincing, and easy to follow.

In keeping with that, I will use my "tests" to determine where I am weak, and what areas I am to focus on more.

I began the book in the beginning (it seemed like a natural place to start), with a section called "techniques of proof". It covered proof by contradiction, proof by induction, etc. (These are all ideas I am familiar with, but the problems they gave were still not easy).

Today, I did some problems from the "'Proof By Contradiction" section, here is one:
(Proof by contradiction is assuming the opposite of what you want to prove, and from there, deriving something that is false, or contradicts your initial assumption)


Every point of three-dimensional space is colored red, green, or blue. Prove that one
of the colors attains all distances, meaning that any positive real number represents
the distance between two points of this color.

Let us assume this is false: That there are positive reals x,y,z such that no two red points are distance x apart, no two blues are distance y apart, and no two greens are distance z apart.


Pick a red point, call the sphere with radius x surrounding it S. All the points in S are blue or green, because otherwise we would have a red point a distance x from the center, which is also red, which would contradict our assumption.

Pick a green point on the sphere. (If there are no green points, all the points on the sphere are blue, so it's easy to find two that are distance y apart)

We find two points on the sphere, Q and R, so PQ=z, PR=z, QR=y. (To see that this is true, anchor a triangle like that to P, and rotate it around P until it's other two points touch the sphere)

If Q or R is green, then it and P make two green points with distance z, which would contradict our initial assumption. So Q and R are both blue, which contradicts our assumption as well, as QR=y, so our assumption is false and our problem is solved.