2002 AMC 10B Problems/Problem 16

Problem

For how many integers $n$ is $\frac{n}{20-n}$ the square of an integer?

For $n=20$ the fraction is undefined, for $n>20$ and $n<0$ it is negative, hence not a square.

For $n=0$ the fraction equals $0$ , which is a square.

For $1\leq n\leq 9$ the fraction is strictly between $0$ and $1$ .

For $n=10$ the fraction equals $1$ , which is a square.

The next square is $4$ , and this is achieved for $n=16$ , and the square after that is $9$ , achieved for $n=18$ .

That leaves $n=19$ , for which the fraction is $19$ , which is not a square.

In total, there are $\boxed{4}$ squares among these fractions.