5. Calculate $f(2)$, if $25 f\left(\frac{x}{1580}\right)+(3-\sqrt{34}) f\left(\frac{1580}{x}\right)=2017 x$. Round the answer to the nearest integer.
Solution: Let's make the substitution: $\mathrm{t}=\frac{x}{1580}$, then the equation will take the form:

$25 f(t)+(3-\sqrt{34}) f\left(\frac{1}{t}\right)=2017 \cdot 1580 \cdot t \quad(1)$,

substitute $\frac{1}{t}$ for $t$ in the equation, we get

$$
25 f\left(\frac{1}{t}\right)+(3-\sqrt{34}) f(t)=2017 \cdot 1580 \cdot \frac{1}{t}
$$

Solve the system of equations

$$
\begin{aligned}
& \left\{\begin{array}{l}
25 f(t)+(3-\sqrt{34}) f\left(\frac{1}{t}\right)=2017 \cdot 1580 \cdot t \\
25 f\left(\frac{1}{t}\right)+(3-\sqrt{34}) f(t)=2017 \cdot 1580 \cdot \frac{1}{t}
\end{array}\right. \\
& \left\{\begin{aligned}
25^{2} f(t)+25(3-\sqrt{34}) f\left(\frac{1}{t}\right) & =25 \cdot 2017 \cdot 1580 \cdot t \\
(3-\sqrt{34})^{2} f(t)+25(3-\sqrt{34}) f\left(\frac{1}{t}\right) & =2017 \cdot 1580 \cdot(3-\sqrt{34}) \cdot \frac{1}{t}^{, \text { then }}
\end{aligned}\right. \\
& \left(25^{2}-(3-\sqrt{34})^{2}\right) f(t)=25 \cdot 2017 \cdot 1580 \cdot t-2017 \cdot 1580 \cdot(3-\sqrt{34}) \cdot \frac{1}{t} \\
& f(t)=\frac{2017 \cdot 1580\left(25 t-(3-\sqrt{34}) \cdot \frac{1}{t}\right)}{\left(25^{2}-(3-\sqrt{34})^{2}\right)}, \text { then } \\
& f(2)=\frac{2017 \cdot 1580\left(50-(3-\sqrt{34}) \cdot \frac{1}{2}\right)}{\left(25^{2}-(3-\sqrt{34})^{2}\right)}=\frac{2017 \cdot 1580 \cdot(97+\sqrt{34})}{2 \cdot(582+6 \sqrt{34})}=\frac{2017 \cdot 1580}{2 \cdot 6} \\
& f(2)=\frac{2017 \cdot 395}{3}=265571 \frac{2}{3} \approx 265572
\end{aligned}
$$

Answer: 265572.