jmc

The harmonic mean of $x$ and $y$ is equal to $\frac{1}{\frac{\frac{1}{x}+\frac{1}{y}}{2}}=\frac{2xy}{x+y}=20$, so we have $xy=10(x+y)$. By Simon's Favorite Factoring Trick, $$xy-10(x+y)+100=(x-10)(y-10)=100.$$Now, $100=2^2\cdot 5^2$ has $(2+1)\cdot (2+1)=9$ factors, or we could simply list all of the possible factors: $\{1,2,4,5,10,20,25,50,100\}$. It follows that there are $\boxed{9}$ possible ordered pairs $(x,y)$.