There are 13 weights, each weighing an integer number of grams. It is known that any 12 of them can be divided into two cups of weights, six weights on each one, which will come to equilibrium. Prove that all the weights have the same weight.
The judges of an Olympiad decided to denote each participant with a natural number in such a way that it would be possible to unambiguously reconstruct the number of points received by each participant in each task, and that from each two participants the one with the greater number would be the participant which received a higher score. Help the judges solve this problem!