For AIME, there are always questions ask m+n or mn where m and n and relatively prime integers. It's easy to infer the only way to m or n = 0 is the other one is 1. The answer of AIME are always integers from 0 to 999.
Suppose m and n are distributed evenly as long as m+n<1000 and mn<1000. How is the distribution of m+n and mn? Because if m and n can't be found easily, an educated guess can increase the chance.

Case 1: For the m+n case, the amount of (m,n) pairs for x are
Case 2: For the mn case, the amount of (m,n) pairs for x are
Where is the Euler's totient function, is the number of unique prime factors for x, I can't find any named function with that definition.

So for m+n one, it's better to chose a larger number, even a prime number.
For mn case, it's better to chose numbers with a lot of unique prime factors.

Case 3: It happens sometimes when m and n doesn't have any other requirement except m+n<1000. Then we have a linear relationship. The amount of times m+n = x is x-1.

Case 4: Sometimes there are Squarefree requirement for n, then it is still approximately linear.

A nice plot. Green = Case 4, Brown = Case 3, Purple = Case 2, Blue = Case 1 * 50
Then one just have to have a random number generator, and then convert it into a pick in the plot.