Balancing numbers

Suppose we have 13 real numbers with the following property: if we remove any one of the numbers, the remaining 12 can be split into two sets of 6 numbers each with equal sum.

Prove: all the 13 numbers are equal.

(Hint: first assume all the numbers are integers.)

http://people.cs.uchicago.edu/~laci/REU07/appuzzles.pdf