Problem 3. Answer: $\frac{1}{64}$.

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Problem 3. Answer: $\frac{1}{64}$.
Solution. The coordinates of each pair of intersection points of the line and the hyperbola satisfy the system of equations:

$$
\left\{\begin{array}{c}
y=\frac{k}{x} \\
y=2 k x+b
\end{array} \rightarrow \frac{k}{x}=2 k x+b \rightarrow 2 k x^{2}+b x-k=0\right.
$$

The product of the abscissas of the two intersection points is $-\frac{1}{2}$, so for twelve points this value is $\frac{1}{2^{6}}=\frac{1}{64}$.