3.1. Functions $f(x)$ and $g(x)$ are defined for all $x>0$; $f(x)$ is the greater of the numbers $x$ and $1 / x$, while $g(x)$ is the lesser of the numbers $x$ and $1 / x$. Solve the equation $f(5 x) \cdot g(8 x) \cdot g(25 x)=1$. If there is only one solution, provide it; if there are multiple solutions, provide their sum. Round the answer to the nearest hundredths if necessary.
Answer: 0.09 (exact value: 0.089).

Solution. The following cases are possible:

a) $0<5 x<8 x<25 x \leqslant 1$. Then $f(5 x) \cdot g(8 x) \cdot g(25 x)=\frac{8 x \cdot 25 x}{5 x}=40 x=1$, from which $x=\frac{1}{40}=0.025$.
b) $0<5 x<8 x \leqslant 1<25 x$. Then $f(5 x) \cdot g(8 x) \cdot g(25 x)=\frac{8 x}{5 x \cdot 25 x}=\frac{8}{125 x}=1$, i.e., $x=\frac{8}{125}=0.064$.
c) $0<5 x \leqslant 1<8 x<25 x$. Then $f(5 x) \cdot g(8 x) \cdot g(25 x)=\frac{1 \cdot 25 x \cdot \frac{1}{5 x \cdot 8 x \cdot 25 x}}{=} \frac{1}{1000 x^{3}}=1$, from which $x=0.1$, which does not satisfy the condition $1<8 x$.
d) $1<5 x<8 x<25 x$. Then $f(5 x) \cdot g(8 x) \cdot g(25 x)=\frac{5 x}{8 x \cdot 25 x}=\frac{1}{40 x}=1$, from which $x=\frac{1}{40}$, which does not satisfy the condition $1<5 x$.

Thus, the equation has two solutions $x=0.025$ and $x=0.064$, the sum of which is $0.089 \approx 0.09$.

Remark. The exact value 0.089 is also accepted as a correct answer.