CONO-SUR 2008 Problem 3

Two friends $A$ and $B$ must solve the following puzzle. Each of them receives a number from the set $\{1,2,…,250\}$ , but they don't see the number that the other received. The objective of each friend is to discover the other friend's number. The procedure is as follows: each friend, by turns, announces various not necessarily distinct positive integers: first $A$ says a number, then $B$ says one, $A$ says a number again, etc., in such a way that the sum of all the numbers said is $20$ . Demonstrate that there exists a strategy that $A$ and $B$ have previously agreed on such that they can reach the objective, no matter which number each one received at the beginning of the puzzle.