The forty-fifth question: Given $\mathrm{n}(\mathrm{n} \geq 4)$ points on a plane, no three points are collinear, and a line segment is drawn between any two points, find the maximum number of line segments that do not intersect with any other line segments (except at the endpoints).

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Question 45:
Answer: A line segment that does not intersect with any other line segments is called a "special line segment".
Below is the proof that the number of special line segments is at most $2n-2$.
When $n=4$, the conclusion is obviously true.
For $n>4$, assume the conclusion holds for $n-1$. Next, consider the case of $n$ points.
If the $n$ points are all vertices of a convex $n$-gon, then the special line segments are only $n$ (i.e., the sides of the convex $n$-gon). Since $n>4$, there is at least one point $Q$ outside $\triangle P_{1} P_{2} P_{3}$, and $AQ$ must intersect at least one of $P_{1}P_{2}$, $P_{2}P_{3}$, or $P_{1}P_{3}$. Therefore, the number of special line segments is at most $2n-4+3-1=2n-2$.
Finally, we provide an example of $2n-2$ special line segments.
As shown in the figure, let $A$ be a point outside a circle, and draw two tangents from point $A$ to the circle, with the points of tangency being $X$ and $Y$. On the minor arc $XY$, take points $A_{1}, A_{2}, \ldots, A_{n-1}$ sequentially. For these $n$ points $A, A_{1}, A_{2}, \ldots, A_{n-1}$, the line segments $AA_{i}, A_{i}A_{i+1}$ are all special line segments, where $i=1, 2, \ldots, n-1$, and let $A_{n}=A_{1}$.