$\underline{\text { Zvonkin }}$ D:

It is required to make a set of weights, each of which weighs an integer number of grams, with which it is possible to weigh any integer weight from 1 to 55 grams inclusive, even in the case where some weights are lost (the weights are placed on one pan of the scales, the weight to be measured - on the other). Consider two versions of the problem:

a) it is necessary to select 10 weights, from which any one can be lost;

b) it is necessary to select 12 weights, from which any two can be lost.
a) The desired set of weights: $f_{1}=1, f_{2}=1, f_{3}=2, f_{4}=3, f_{5}=5, f_{6}=8, f_{7}=13, f_{8}=21, f_{9}=34, f_{10}=55$ (terms of the Fibonacci sequence, for which $\left.f_{n+2}=f_{n+1}+f_{n}\right)$.

By induction on $n$, we will prove that even after losing one weight from the set of weights $f_{1}, f_{2}, \ldots, f_{n}$, any weight from 1 to $f_{n+1}-1$ can be composed. The base case is obvious.

Inductive step. If the weight $S<f_{n}$, then by the inductive hypothesis, it can be weighed using the set $f_{1}, f_{2}, \ldots, f_{n-1}$ with one weight lost. Suppose $S \geq f_{n}$. If $f_{n}$ is not lost, then the weight $S-f_{n}$ can be weighed using the weights $f_{1}, f_{2}, \ldots, f_{n-1}$ (with the lost weight) and adding the weight $f_{n}$. If $f_{n}$ is lost, then $f_{n-1}$ is not lost. Subtracting it from $S$, we get a weight less than $f_{n+1}-f_{n-1}=f_{n}$. This can be weighed using the set $f_{1}, f_{2}, \ldots, f_{n-1}$ with the lost weight $f_{n-1}$. Adding $f_{n-1}$, we get $S$.

b) The desired set of weights $F_{1}=F_{2}=F_{3}=1, F_{4}=2, F_{5}=3, F_{6}=4, F_{7}=6, F_{8}=9, F_{9}=13, F_{10}=19, F_{11}=28$, $F_{12}=41$ (terms of the sequence for which $F_{n+1}=F_{n}+F_{n-2}$). Similarly to part a), by induction, we will prove that after losing two weights from the set $F_{1}, \ldots, F_{n}$, any weight $S<F_{n+1}$ can be weighed. In the inductive step, if the weight $F_{n}$ is lost, then one of the weights $F_{n-1}, F_{n-2}$ is not lost. Subtracting it from $S$, we get a weight less than $F_{n+1}-F_{n-2}=F_{n}$, which can be weighed using the set of weights $F_{1}, \ldots, F_{n-1}$, and the previously used weight should be considered lost.

Thus, the presented set allows (with the loss of two weights) to weigh any integer weight from 1 to $41+$ $19-1=59$ grams.