Math Contest Questions

bjgawp

Hi everyone. Getting to the point, the types of questions that math contests have are not really taught, so to speak, in high school. I was just wondering if anyone had any resources or tips for solving these kinds of questions. For example:
Suppose n and D are integers with n positive and 0 ≤ D ≤ 9. Determine n if n / 810 = 0.9D59D59D5 ...
- Questions that involve repeating decimals and such ... I have no clue where to start

Four tiles identical to the one shown, with a > b > 0, are arranged without overlap to form a square with a square hole in the middle.

Determine all positive integers N for which there are odd integers a > b > 0 such that the ratio of the area of the inner square to the area of the outer square is 1:N.
- Geometric problems involving ratios are often confusing as well.

I don't expect anyone to answer the questions (I attempted these questions at a math contest already). I just need some tips for future questions that may ressemble these.

Gokul43201

Most number theory questions, like the first one above, are actually accessible through stuff taught in school. For the problem above, you can use the standard technique that's used for converting a repeating decimal to a rational fraction. Beyond that point, it's just being clever about numbers that are divisible by 11. I've found that for many of these problems you develop a good idea of how to solve them as well as the ability to find solutions quickly, if you learn a little bit of modulo arithmetic. This is learnable by anyone with a high school level math background. You will find a chapter on modulo arithmetic in any introductory text on number theory (e.g., Burton).