Combinatorics - sending letters 
In a group of 20 people, everybody sends a letter to 10 different people (not to himself). Are there always two people who send each other a letter?

So I'm stuck on this neat little problem. I tried looking at simpler cases, where in a group of $2n$ people, everybody sends a letter to $n$ different people. Just from fiddling around with it I reckon it isn't possible without two people sending each other a letter. 
Can anyone provide help with this problem?

Draw a directed graph where an arrow from a to b means a sent letter to b. The average in degree is the average out degree and this is 10. So at least one person P get letters from 10 different people and so one of the persons sending a letter to P also get a letter from P.
