Task 1. We call a collection of lines in the plane beautiful if each line in the collection intersects an odd number of the other lines in the collection.

Determine the smallest integer $k \geq 0$ with the following property: for any 2018 different lines $\ell_{1}, \ell_{2}, \ldots, \ell_{2018}$ in the plane, there exist lines $\ell_{2018+1}, \ell_{2018+2}, \ldots$, $\ell_{2018+k}$ such that the lines $\ell_{1}, \ell_{2}, \ldots, \ell_{2018+k}$ are all different and form a beautiful collection.
Solution. First, we prove that in a beautiful set, the number of lines must be even. Assume that the number of lines is odd. Then, on each of the odd number of lines, there lies an odd number of intersection points, so the total number of intersection points is also odd. However, each intersection point is counted twice (once for each line it lies on), so the total should be even. Contradiction. Therefore, the number of lines in a beautiful set must be even. In particular, $2018+k$ must also be even, so $k$ must be even.

Now, assume there are 1009 directions such that in each direction, two of the original lines lie. Then each line is parallel to exactly one other line. If $k<1010$, then $k \leq 1008$, so there must be a direction in which we do not add a line. Consider an original line $\ell$ in this direction. It intersects all lines except itself and the line it is parallel to. This is an even number. Therefore, the resulting set is not beautiful. Thus, there is an example where at least 1010 lines are needed. We now show that it is possible to add exactly 1010 lines so that the total set is ultimately beautiful. We consider all directions in which an even number (greater than 0) of the original lines lie. There are at most 1009 such directions. For each of these directions, we add a line. Each line is then parallel to an even number (possibly 0) of other lines. First, assume that the total number of lines is now even. Then each line intersects an odd number of lines (the total minus itself and an even number of other lines). Now, assume that the total number of lines is now odd. Then each line intersects an even number of lines. We add a line in a new direction that thus intersects all lines (an odd number), so that afterwards each line intersects an odd number of other lines.

$\mathrm{Now}$ the set is beautiful and we have added at most $1009+1=1010$ lines. It is possible that fewer than 1010 lines were added. In that case, the number of added lines is even, because the total number of lines in a beautiful set is always even. We choose a direction in which at least one line already lies. We now add pairs of lines in that same direction. These lines intersect all lines in other directions; that is a total of an odd number. The other lines each gain two intersection points,
so they still have an odd number of intersection points. This also applies to the lines that were already in the chosen direction, as they gain zero new intersection points. The set remains beautiful. We keep adding such pairs until we have added a total of 1010 lines.

We conclude that the minimal $k$ that satisfies the condition is $k=1010$.