# Problem 1.

It is known that the numbers $\frac{x}{2}, 2 x-3, \frac{18}{x}+1$, taken in the given order, form a geometric progression. Find the common ratio of this progression. Round your answer to two decimal places.
Answer: $\frac{52}{25}=2.08$

Solution. By the characteristic property of a geometric progression $a_{n} \cdot a_{n+2}=a_{n+1}^{2}$. We get the equation: $\frac{x}{2} \cdot\left(\frac{18}{x}+1\right)=(2 x-3)^{2}$. Hence $8 x^{2}-25 x=0$. At $x=0$ the domain is violated, and at $x=\frac{25}{8}$ we get a geometric progression with the first term $a=\frac{25}{16}$ and the common ratio $q=\frac{52}{25}$.

## B-2

It is known that the numbers $\frac{x}{2}, 3 x-2, \frac{8}{x}+1$, taken in the given order, form a geometric progression. Find the common ratio of this progression. Round the answer to two decimal places.

Answer: 3.12

## B-3

It is known that the numbers $\frac{x}{2}, 2 x-5, \frac{50}{x}+3$, taken in the given order, form a geometric progression. Find the common ratio of this progression. Round the answer to two decimal places.

Answer: $\frac{92}{43} \approx 2.14$

## B-4

It is known that the numbers $\frac{x}{2}, 3 x-5, \frac{50}{x}+3$, taken in the given order, form a geometric progression. Find the common ratio of this progression. Round the answer to two decimal places.

$$
\text { Answer: } \frac{22}{7} \approx 3.14
$$

B-5 The number A, written in a natural base n, is equal to 14.6, and in the decimal system it is equal to 16.5. The number B in the base n system is equal to 19.3. What is it equal to in the decimal system?

Answer: 21.25

Solution. The equation $14.6_{n}=16.5_{10}$ is equivalent to the equation $n+4+\frac{6}{n}=16.5 \Leftrightarrow 2 n^{2}-25 n+$ $12=0$, which has roots 12 and 0.5. Therefore, $n=12$, so $B=19.3_{12}=1 \cdot 12+9 \cdot 1+\frac{3}{12}=21.25$.

B-6 The number A, written in a natural base n, is equal to 15.4, and in the decimal system it is equal to 21.25. The number B in the base n system is equal to 27.8. What is it equal to in the decimal system?

Answer: 39.5

B-7 The number A, written in a natural base n, is equal to 17.3, and in the decimal system it is equal to 19.25. The number B in the base n system is equal to 22.6. What is it equal to in the decimal system?

Answer: 26.5

B-8 The number A, written in a natural base n, is equal to 14.8, and in the decimal system it is equal to 20.5. The number B in the base n system is equal to 24.4. What is it equal to in the decimal system?

Answer: 36.25

## Lomonosov Olympiad for Schoolchildren in Mathematics

Preliminary stage 2020/21 academic year for $10-11$ grades

## B-1

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