## Task 4 - 260834

In a plane, there are 100 different points arranged such that the following conditions are met:

(1) There is exactly one line on which more than two of the 100 points lie.

(2) On this line, exactly three of the 100 points lie.

Determine, under these conditions, the number of lines that pass through more than one of the 100 points!
The three points mentioned in (2), which lie on the line mentioned in (1), are $P_{98}, P_{99}, P_{100}$, and the remaining 97 of the 100 points are called $P_{1}, P_{2}, \ldots, P_{97}$.

A line passes through more than one of the 100 points if and only if it

(a) connects a pair of different points $P_{1}, P_{2}, \ldots, P_{97}$

(b) or connects one of the points $P_{1}, P_{2}, \ldots, P_{97}$ with one of the points $P_{98}, P_{99}, P_{100}$

(c) or the line is the one through $P_{98}, P_{99}, P_{100}$.

Each line mentioned in (a) is different from each line mentioned in (b). Furthermore, each line mentioned in (a) and each line mentioned in (b) is different from the line mentioned in (c).

The number of lines mentioned in (a) can be determined as follows:

Each of the 97 points $P_{1}, P_{2}, \ldots, P_{97}$ can be connected to each of the 96 other points among these points by a line. In this way, each line mentioned in (a) is counted exactly twice. Therefore, their number is

$$
\frac{97 \cdot 96}{2}=97 \cdot 48
$$

The number of lines mentioned in (b) is $97 \cdot 3$. Thus, the desired number is $97 \cdot 48 + 97 \cdot 3 + 1 = 4948$.

Hint: One can also proceed with a step-by-step consideration:

For 3 points that satisfy (1) and (2) (with the number 3 instead of 100), there is exactly one desired line. Each additional point that is added in such a way that (1) and (2) (with the new number $n$ of points) are still satisfied, adds exactly $n-1$ new lines. This leads to the desired number $1 + 3 + 4 + \ldots + 98 + 99 = 1 + (3 + 99) + (4 + 98) + \ldots + (50 + 52) + 51 = 1 + 48 \cdot 102 + 51 = 4948$.