What's a good book for a beginner in high school math competitions?

It sounds like you want some of the Art of Problem Solving books.

I personally got a lot out of Engel's Problem Solving Strategies in high school. People also seem to like Zeitz's The Art and Craft of Problem Solving, or at least they used to back in the day, but I haven't read it myself.

I think that if you have time, the best you can do is find a bank of problems somewhere, try to attack them all and give yourself the solution if you feel the problem is out of your range. Each time you read a solution, try not only to understand it, but try to understand from what angle should one have attacked the problem in order to obtain the solution. That will not only show you new proof techniques, but also new ways to think about a problem whose solution you don't have.

Unfortunately I don't know any such book as you describe it.

That website you linked (brilliant) looks brilliant! You should try to master all the identities and techniques described there if you plan on taking on a contest, I believe they're all going to be useful. That's the best I can say.

Hope that helps,