RO-TST 2003 Problem 6

At a math contest there are $2n$ students participating. Each of them submits a problem to the jury, which thereafter gives each students one of the $2n$ problems submitted. One says that the contest is fair is there are $n$ participants which receive their problems from the other $n$ participants.

Prove that the number of distributions of the problems in order to obtain a fair contest is a perfect square.