The key to this sort of problem is to be systematic, and Andrei explained very clearly how he worked through every possibility. Here is his solution:
I started from the observation that I must add two 3-digit numbers, and the result is 4-digit one, so that I must assign the value 1 to the letter F.
My method is to give values to the letters from right to left, the way additions are performed. [O is also a good place to start as it occurs three times. - Ed.]
So, the only solutions are: