While learning different formulas and methods is useful for math contests, learning how to problem solve is just as important.

The following strategies apply generally and some will be specific to math contests at the level of the AMC's.

It's usually helpful to think about the edge or extreme case of a problem. By doing this, you can check if it's possible, and if it's not,

What is the maximum number of queens you can place on squares in a $8 \times 8$ chessboard such that each row, each column and each long diagonal contains at most $4$ queens?

When solving for a general variable, it can be useful to test small test cases and then make a conjecture about the rest of the problem.

Suppose $n$ is a prime number where $n^2 + 11$ has exactly six different positive divisors (including $1$ and the number itself). What is the sum of all possible values of $n$?

In some situations, especially in multiple-choice problems, systematically eliminating impossible answers can be a more efficient strategy than directly attempting to find the correct answer. This technique is useful when the problem seems too complex to solve directly in the given time. By eliminating options that are clearly wrong based on logical reasoning or partial calculations, you can increase your chances of selecting the correct answer from the remaining options.

Practice Problem

For all positive integers $n$, an expression is always divisible by all of the following positive integers except one. Which one is it not divisible by?

The numbers are: $2, 3, 6, 10, 20.$

Casework, or Case checking is a useful strategy of breaking down a complex problem into smaller problems (cases), and then combining them to get the full solution.

Find the sum of all solutions to the equation