Fedorov R.M.

The numerator and denominator of the fraction are natural numbers that sum to 101. It is known that the fraction does not exceed $1 / 3$.

Indicate the greatest possible value of such a fraction.
The sum of the numerator and denominator is 101. Therefore, the larger the numerator of the fraction, the smaller its denominator - and the larger the fraction itself (since both the numerator and the denominator are positive numbers). ${ }^{25} / 76$ is even smaller than ${ }^{1 / 3}, \mathrm{a}^{26} / 75$ - is already larger.

## Answer

$25 / 76$.

Authors: Blyunkov A.d:, Khachatryan A.V.

Solve the puzzle: BAO $\times$ BA $\times 5=2002$.

## Hint

$2002=2 \cdot 7 \cdot 11 \cdot 13$

## Solution

Note that B = 1. Indeed, if B $\geq 2$, then BAO $\times$ BA $\times$ B $\geq 200 \cdot 20 \cdot 2=8000>2002$.

$2002=2 \cdot 7 \cdot 11 \cdot 13$. BA is a two-digit divisor of the number 2002, starting with the digit 1, so BA can be 11, 13, or $2 \cdot 7=14$. Since $B \neq$ A, then $BA \neq 11$. If BA $=13$, then BAO $=2002: 13=154$, from which A is both 3 and 5. Contradiction.

The remaining option BA $=14$, BAO $=2002: 14=143$ is the solution.

## Answer

$143 \cdot 14 \cdot 1=2002$.