## Car: Shapovov A.B.

Vasya wrote a correct statement:

"In this phrase $1 / 3$ of all digits are the digit 3, and $1 / 2$ of all digits are the digit $1$."

And Kolya wrote the phrase:

"In this phrase 1/... of all digits are the digit *, the fractions of digits * and * are equal and are 1/..., while the fraction of all other digits is $1 / \ldots$."

Insert three different digits in place of the asterisks, and three different numbers in place of the ellipses so that the statement is correct.
If any of the numbers replaced by ellipsis is at least three digits, then there are no fewer than 100 digits in this phrase, which is impossible (then one of the numbers would consist of at least 30 digits, but then the total number of digits would be no less than $10^{30}$, and so on, it is clear that this cannot be). Therefore, all numbers are either single-digit or double-digit, and the total number of digits is between 9 and 12.

The number of "1"s is no less than 4, their proportion is no less than $4 / 12 = 1 / 3$, so the denominator of the first fraction is a single-digit number, meaning there are fewer than 12 digits.

All denominators are divisors of the number of digits, greater than one. Therefore, the number of digits cannot be 11 (the number 11 has no single-digit divisors other than one), nor 9 (four terms of the form $1 / 3$ and $1 / 9$ cannot sum to 1). Thus, the total number of digits is 10, and the proportion of "1"s is $1 / 2$. The other fractions can be $1 / 5$ and $1 / 10$. The sum of the proportions must equal $1: 1 / 2 + 1 / 5 + 1 / 5 + 1 / 10$.

The denominators include the digits 0, 2, and 5 once each. Any two can be mentioned explicitly, then their proportion will be 11/5, and the proportion of the remaining single digit will be $\frac{1}{10} 10$.

## Answer

"In this phrase, 1/2 of all digits are the digit 1, the proportions of the digits 2 and 5 are the same and equal to 1/5, and the proportion of all other digits is $1 / 10$

(or "...the proportions of the digits 0 and 2 are the same..." or "...the proportions of the digits 0 and 5 are the same...").

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