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?

There were $200$ letters sent in total. So at least one person must have received at least $10$ letters (because if everyone got at most $9$ letters, then only at most $180$ letters were received).
Say that Bob received at least $10$ letters. Since there are $19$ people who are not Bob, there are at most $9$ other people who didn't send a letter to Bob. Since Bob sent $10$ letters, he must have sent a letter to one of the people who sent one to him.
