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jobsism

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However, this post is on a completely optimistic note; after looking at the ingenuity of the questions asked, and the fun in the math involved to solve them, I'm well damn determined to go to whatever extent to learn the math to solve such problems! :D

All I want to know now is whether my current preparation is sufficient; I'm currently studying from a book named "Mathematical Circles: A Russian Experience" by Dmitri Fomin, Sergey Genkin and Ilia Itenberg. The theory is easy to understand and the problems are intriguing. However, I'm still not at a level to solve the "high-level" ones (perhaps I will be able to, after completing the book). And I think the theory is a bit limited too. To have an idea of these "high-level" (atleast for me!) questions, here's a sample:-

1. Show that there are exactly 16 pairs of integers (x, y) such that 11x + 8y + 17 = xy.

2. A function g from a set X to itself satisfies g^m = g^n for positive integers m and n with m > n. Here g^n stands for g ◦ g ◦ · · · ◦ g (n times). Show that g is one-to-one if and only if g is onto.

3. Let a1, a2,...a100 be 100 positive integers. Show that for some m,n with 1<=m<=n<=100, ∑[i=m to n] a(subscript)i is divisible by 100.

4. In Triangle ABC, BE is a median, and O the mid-point of BE. The line joining A and O meets BC at D. Find the ratio AO : OD.

Are there any other beginner-level books, that cover most topics? I've already glanced at some of the famous ones like "Mathematical Olympiad Challenges" by Titu Andreescu, "The Art and Craft of Problem-Solving" by Paul Zeitz and "Problem-Solving Strategies" by Arthur Engel, but I don't feel comfortable using them.

Any other book suggestions/tips, anyone? Please help.

Thanks! :D