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

$S_0=290797$
$S_{n+1}=S^2_n \mod 50515093$
$T_n=(S_n \mod 2000) - 1000$

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

$S_0=290797$
$S_{n+1}=S^2_n \mod 50515093$
$T_n=(S_n \mod 2000) - 1000$