9.4. From Zlatoust to Miass, a "GAZ", a "MAZ", and a "KAMAZ" set off simultaneously. The "KAMAZ", having reached Miass, immediately turned back and met the "MAZ" 18 km from Miass, and the "GAZ" - 25 km from Miass. The "MAZ", having reached Miass, also immediately turned back and met the "GAZ" 8 km from Miass. What is the distance from Zlatoust to Miass?
Answer: 60 km.

Solution. Let the distance between the cities be $x$ km, and the speeds of the trucks: "GAZ" $-g$ km/h, "MAZ" - $m$ km/h, "KAMAZ" $-k$ km/h. For each pair of vehicles, we equate their travel time until they meet. We get $\frac{x+18}{k}=\frac{x-18}{m}, \frac{x+25}{k}=\frac{x-25}{g}$ and $\frac{x+8}{m}=\frac{x-8}{g}$. Let's rewrite these equations differently: $\frac{x+18}{x-18}=\frac{k}{m}, \frac{x-25}{x+25}=\frac{g}{k}$ and $\frac{x+8}{x-8}=\frac{m}{g}$. Multiplying them term by term, we get: $\frac{x+18}{x-18} \cdot \frac{x-25}{x+25} \cdot \frac{x+8}{x-8}=1$.

Transform the obtained equation, considering that the denominator of each fraction is not zero: $x^{3}+x^{2}+(18 \cdot 8-18 \cdot 25-8 \cdot 25) x-18 \cdot 8 \cdot 25=x^{3}-x^{2}+(18 \cdot 8-18 \cdot 25-8 \cdot 25) x+18 \cdot 8 \cdot 25 \Leftrightarrow 2 x^{2}=2 \cdot 18 \cdot 8 \cdot 25$. Since $x>0$, then $x=60$.

## Grading Criteria:

+ a complete and well-reasoned solution is provided

$\pm$ the equation with one variable (the unknown) is correctly derived, but an error is made in solving it

干 the three proportions are correctly written, but there is no further progress

$\mp$ only the correct answer is provided

- the problem is not solved or is solved incorrectly