AMC12 2004 12A, Problem 3

For how many ordered pairs of positive integers $(x,y)$ is $x + 2y = 100$?

Solution: If $x$ and $2y$ must each be positive integers, then we can say that $x$ is at least 1 and $2y$ is at least 1. From there, we want to find out how many ways there are to distribute the other 98 ones (the smallest positive integer addends of 100). 98 identical objects can be distributed to two distinct bins in 99 ways (think stars and bars), yet this 99 is an overcount. Because $y$ must be an integer, $2y$ must be even; thus only $\left\lfloor \frac{99}{2} \right\rfloor = \fbox{49}$ ways exist to distribute these ones.

Answer: 49