# Problem 5.

B-1

Numbers $x, y, z$ are such that $\frac{x+\frac{53}{18} y-\frac{143}{9} z}{z}=\frac{\frac{3}{8} x-\frac{17}{4} y+z}{y}=1$. Find $\frac{y}{z}$.
Answer: $\frac{352}{305} \approx 1.15$

Solution. We will solve the problem in a general form: find $\frac{y}{z}$, if the numbers $x, y, z$ are such that $\frac{x+A y-B z}{z}=\frac{C x-D y+z}{y}=1$. We will use the fact that if $\frac{a}{b}=\frac{c}{d}=k$, then $\frac{a+n c}{b+n d}=k$ for any $n$ for which the denominator does not become zero. Then

$$
\begin{aligned}
1= & \frac{C x-D y+z-C x-C A y+C B z}{y-C z}=\frac{-(D+C A) y+(C B+1) z}{y-C z}=\frac{-(D+C A) \frac{y}{z}+(C B+1)}{\frac{y}{z}-C} \\
& \frac{y}{z}-C=-(D+C A) \frac{y}{z}+(C B+1) ; \quad \frac{y}{z}=\frac{C+C B+1}{D+C A+1}
\end{aligned}
$$

## B-2

The numbers $x, y, z$ are such that $\frac{x+\frac{53}{18} y-\frac{157}{9} z}{z}=\frac{\frac{3}{8} x-\frac{17}{4} y+z}{y}=1$. Find $\frac{y}{z}$. Round the answer to the hundredths. Answer: $\frac{76}{61} \approx 1.25$

B-3

The numbers $x, y, z$ are such that $\frac{y+\frac{53}{18} z-\frac{121}{2} x}{x}=\frac{\frac{3}{8} y-\frac{4}{3} z+x}{z}=1$. Find $\frac{z}{x}$.

Answer: 7

## B-4

The numbers $x, y, z$ are such that $\frac{z+\frac{53}{18} x-\frac{55}{2} y}{y}=\frac{\frac{3}{8} z-\frac{4}{3} x+y}{x}=1$. Find $\frac{x}{y}$.

Answer: 3.4

## Lomonosov Olympiad for Schoolchildren in Mathematics

Preliminary Stage 2020/21 Academic Year for 9th Grade