Sometimes, the hardest part of a problem is not the math itself--it's translating from English into math! Mathcounts usually includes lots of word problems, so you should have a secure grasp on this skill.
Generally, start by identifying what your variables are going to represent. Ask yourself, What is unknown in this problem? The answer to that question usually shows you what the variables will stand for. Once you define your variables, it's a good idea to jot down what they stand for. "x = cows, y = $" is sufficient. You don't want to solve the entire system of equations, only to realize that you've forgotten what x and y mean!
Next, identify the value you are seeking. What does the problem ask for? Which variable(s) do you need to solve for? Sometimes, you will simply need to provide the value of one of your variables. Other times, you may need to do more work after solving the equations, or you may have to combine the values of your variables. Always work with the goal in mind; it can save you a lot of unnecessary steps, and it will help prevent the "stupid mistake" of writing the wrong value in the answer blank.
Now it's time to write the equations. Look for values that are equal to one another. The word "is" is a big keyword that often means "equals." Some more keywords that can help you write equations:
Once you've written your equations, solving them should be more or less straightforward, assuming that you've written them correctly. If the number of equations is equal to (or more than) the number of variables, then the system should be solvable. If there are more variables than equations, then you will not be able to solve for all variables. However, you may still be able to discover valuable information, such as the ratio between variables.