C3 (6-5, Romania) On the plane, given 5 points, the lines connecting these points pairwise are neither parallel, nor perpendicular, nor coincident. From each point, draw perpendiculars to the lines formed by connecting the other four points pairwise. How many intersection points (excluding the given 5 points) can these perpendiculars have at most?
Let $P_{1}, P_{2}, P_{3}, P_{4}, P_{5}$ be the given 5 points. The number of lines connecting any two points is $\mathrm{C}_{5}^{2}=10$, and the number of lines connecting any two points among 4 points is $\mathrm{C}_{4}^{2}=6$. Thus, from one point, 6 perpendicular lines can be drawn, and from 5 points, a total of $5 \times 6=30$ perpendicular lines can be drawn, which can have at most $\mathrm{C}_{30}^{2}=435$ intersection points. Below, we point out that some of these intersection points are the same.

Consider the line $P_{1} P_{2}$, and $P_{3}, P_{4}, P_{5}$ are not on this line. The perpendiculars from $P_{3}, P_{4}, P_{5}$ to $P_{1} P_{2}$ are parallel and do not intersect, so we should subtract $\mathrm{C}_{5}^{2} \cdot \mathrm{C}_{3}^{2} = 30$ intersection points from the total.

Furthermore, from each of the points $P_{1}, P_{2}, \cdots, P_{5}$, 6 perpendiculars can be drawn to the 6 lines determined by the other 4 points. Thus, these 6 lines intersect at one point, and for each of the 5 points, the total loses $\mathrm{C}_{6}^{2} = 15$ points, so we should subtract $5 \times 15 = 75$ points from the total.

Additionally, every three points form a triangle, and there are $\mathrm{C}_{5}^{3}=10$ triangles. Each triangle's three altitudes intersect at one point, so the 10 triangles reduce the total by $10 \times 2 = 20$ intersection points.

The lost intersection points considered above belong to different types and are not repeated, so the maximum number of intersection points is
$$
\begin{aligned}
& \mathrm{C}_{30}^{2}-\mathrm{C}_{5}^{2} \mathrm{C}_{3}^{2}-5 \mathrm{C}_{6}^{2}-\mathrm{C}_{5}^{3}\left(\mathrm{C}_{3}^{2}-1\right) \\
= & 435-30-75-20 \\
= & 310 \text { (points). }
\end{aligned}
$$