Book recommendations for these types of math? I'm planning to write a math olympiad in a couple of months (4-5), and am just really trying to get the preparation in. I'm a fairly good math student (did ok in math, not an A+, but I got an A so my fundamentals are good I think). I was just wondering if someone could suggest some excellent books that are good for beginners and will open my mind to questions involving concepts such as:
-Telescoping sums (If $a_1a_2\cdots a_n=1$, then the sum $\sum_k a_k\prod_{j\le k} (1+a_j)^{-1}$ is bounded below by $1-2^{-n}$)
-Proving arithmetic progressions that meet some sort of criteria think these kinds of questions (Proving a set of numbers has arithmetic progressions of arbitrary length, but none infinite)
etc... and in general constructing proofs about sequences/series, that sort of thing.
That would be really awesome. I'm kind of scrambling all over the place to find the approximate syllabus for these Olympiad questions, so guidance in finding resources to learn would be great.
Thank you.
 A: Which olympiad is it? 
All olympiads have become highly competitive. If you want to do well, then (1) you need to spend a lot of time practising, (2) you need to practise against a time limit. There are large numbers of olympiad problem books available now. And there are lots of problems available on the web. 
But don't turn to the solutions too quickly. A good rule of thumb is that you need to struggle with one hard problem for at least half-an-hour every couple of days. It is no good getting into the habit of giving up and asking for help after 2 minutes.
You will be up against people who will be practising olympiad type problems for dozens of hours every week.
If you want more specific recommendations, I need more details about the particular olympiad and some feel for how good you are. For example, here is an old and easy IMO problem: A is the sum of the decimal digits of $4444^{4444}$ and B is the sum of the decimal digits of A. Find the sum of the decimal digits of B. 
How long did it take you to do that?
