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 1 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 to problem $\underline{98056}$), and it only factors into two factors: $201=3 \cdot 67$.

## Answer

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

b) 3 sets.

Author: Fomin D:

Given 103 coins of the same appearance. It is known that two of them are counterfeit, that all genuine coins weigh the same, and that the counterfeit coins also weigh the same but differ from the weight of the genuine coins. However, it is unknown whether the counterfeit coins are heavier or lighter than the genuine ones. How can this be determined using three weighings on a two-pan balance without weights? (It is not required to separate the counterfeit coins.)

## Solution

Suppose we have $6 k+1$ coins, two of which are counterfeit. Discard one coin, and divide the rest into three piles $-A, B, C$ - each containing $2 k$ coins. Note that among them, there will be exactly two piles with the same weight. Weigh $A$ against $B$ first, and then $B$ against $C$. Suppose, for example, that $A$ is heavier than $B$ and $C$. Then $B$ contains no more than one counterfeit coin (since the weights of $B$ and $C$ are equal). Divide $B$ into two equal parts and weigh them. If the weights are equal, then the counterfeit coin is heavier than the genuine one, otherwise, it is lighter.