# Task 4.

In modern conditions, digitalization - the conversion of all information into digital code - is considered relevant. Each letter of the alphabet can be assigned a non-negative integer, called the code of the letter. Then, the weight of a word can be defined as the sum of the codes of all the letters in that word. Is it possible to encode the letters О, П, С, Т, Ь, Я with elementary codes, each consisting of a single digit from 0 to 9, so that the weight of the word "СТО" is not less than the weight of the word "ПЯТЬСОТ"? If such encoding is possible, in how many ways can it be implemented? If such encoding is possible, does it allow for the unambiguous restoration of a word from its code?
# Solution.

Let $k(x)$ denote the elementary code of the letter $x$. We have:

$$
k(C)+k(T)+k(O) \geq k(\Pi)+k(\text { Ya) }+k(T)+k(\text{ b })+k(C)+k(O)+k(T)
$$

which is equivalent to

$$
k(\Pi)+k(T)+k(\text{ b })+k(\text { Ya) }=0
$$

from which it follows that

$$
k(\Pi)=k(T)=k(\text{ b })=k(\text { Ya) }=0
$$

The elementary codes of the two remaining letters $k(C)$ and $k(O)$ can be any digits from 0 to 9. Therefore, there are exactly $10^{2}$ ways to encode them. However, for any choice of $k(O)$ and $k(C)$, the words "POP" and "TOT" will have the same codes $k(O)$, which cannot be decoded uniquely.

Answer. The inequality for the weights holds if and only if $k(\Pi)=k(T)=k(\text{ b })=k(\text { Ya) }=0$.

There are exactly 100 ways to encode them.

In any of these ways, unique decoding is impossible.