Problem 630

Crossed lines

Given a set, $L$ , of unique lines, let $M (L)$ be the number of lines in the set and let $S (L)$ be the sum over every line of the number of times that line is crossed by another line in the set. For example, two sets of three lines are shown below:

In both cases $M (L)$ is $3$ and $S (L)$ is $6$ : each of the three lines is crossed by two other lines. Note that even if the lines cross at a single point, all of the separate crossings of lines are counted.

Consider points $(T_{2 k - 1}, T_{2 k})$ , for integer $k \geq 1$ , generated in the following way:

$S_{0} = 290797$
$S_{n + 1} = S_{n}^{2} \mod 50515093$
$T_{n} = (S_{n} \mod 2000) - 1000$

For example, the first three points are: (527, 144), (−488, 732), (−454, −947). Given the first $n$ points generated in this manner, let $L_{n}$ be the set of unique lines that can be formed by joining each point with every other point, the lines being extended indefinitely in both directions. We can then define $M (L_{n})$ and $S (L_{n})$ as described above.

For example, $M (L_{3}) = 3$ and $S (L_{3}) = 6$ . Also $M (L_{100}) = 4948$ and $S (L_{100}) = 24477690$ .

Find $S (L_{2500})$ .

交叉的直线

给定一系列相异直线的集合 $L$ ，记 $M (L)$ 为集合中直线的数目，而 $S (L)$ 为所有这些直线与集合中其它直线相交的次数之和。例如，考虑下图所示的两组直线集合：

在这两种情形中， $M (L)$ 均为 $3$ 而 $S (L)$ 均为 $6$ ：每个集合有三条直线，每条直线都与其它两条直线各相交一次。注意到，即使这些直线交于同一点，每次相交也都分别计算。

考虑由以下方式构造的一系列点 $(T_{2 k - 1}, T_{2 k})$ ，其中整数 $k \geq 1$ ：

$S_{0} = 290797$
$S_{n + 1} = S_{n}^{2} \mod 50515093$
$T_{n} = (S_{n} \mod 2000) - 1000$

例如，前三个点分别是：(527, 144)，(−488, 732)，(−454, −947)。给定由这种方式给出的前 $n$ 个点，并记 $L_{n}$ 为将这些点两两连接所得的所有相异直线构成的集合；注意直线总是向两端延长至无限远处。相应地，可以按照上述定义给出 $M (L_{n})$ 和 $S (L_{n})$ 。

例如， $M (L_{3}) = 3$ 而 $S (L_{3}) = 6$ 。此外， $M (L_{100}) = 4948$ 而 $S (L_{100}) = 24477690$ 。

求 $S (L_{2500})$ 。