Fomin d:

Consider a set of weights, each weighing an integer number of grams, and the total weight of all weights is 200 grams. Such a set is called correct if any body with a weight expressed as an integer number of grams from 1 to 200 can be balanced by some number of weights from the set, and in a unique way (the body is placed on one pan of the scales, the weights - on the other; two balancing methods that differ only by the replacement of some weights with others of the same weight are considered the same).

a) Provide an example of a correct set where not all weights are one gram each.

b) How many different correct sets exist?

(Two sets are different if some weight participates in these sets a different number of times.)
The correct set should correspond to the factorization of the number 201 (see the solution of problem $\underline{98056}$), and it only factors into two factors: $201=3 \cdot 67$.

## Answer

a) Two weights of 67 g and 66 weights of 1 g or 66 weights of 3 g and two of 1 g.

b) 3 sets.