9,10,11 |
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|  | [ Lines, rays, segments, and angles (miscellaneous). |  |
|  | [ Examples and counterexamples. Constructions $]$ |  |

On a plane, $n>2$ lines in general position are drawn (that is, no two lines are parallel and no three lines intersect at the same point). These lines have divided the plane into several parts. What is the

a) smallest;

b) largest

number of angles that can be among these parts?
a) Consider the polygon $T$, which is the union of all bounded parts. It is clear that all angles are perpendicular to the angles of $T$ that are less than $180^{\circ}$. From the formula for the sum of exterior angles, it immediately follows that there are no fewer than three such angles.

An example with three angles for $n>3$ can be constructed as follows. Take a point $D$ inside the triangle $ABC$, inscribe a circle of sufficiently small radius in the angle $ADB$, take $n-4$ points on the smaller arc formed by the points of tangency, and draw tangents at these points. These tangents, together with the lines $AC, BC, AD, BD$, form the desired set.

b) Construct a circle that contains all the points of intersection. These lines divide it into $2n$ arcs. Let $AB, BC$ be two adjacent arcs, and $X, Y$ be the points of intersection of the line passing through $B$ with the lines passing through $A$ and $C$. Then, if $X$ lies on the segment $BY$, the part of the plane containing the arc $BC$ is not an angle, that is, of the two parts containing adjacent arcs, only one can be an angle. Therefore, the number of angles does not exceed $n$, and equality is possible only when each second arc is part of an angle. But for even $n$, this means that there are two angles containing opposite arcs, that is, formed by the same pair of lines. This is obviously impossible. For odd $n$, the lines containing the sides of a regular $n$-gon divide the plane into parts, $n$ of which are angles. It is obvious that one more line can be added to them so that the number of angles does not decrease.

## Answer

a) 3 ; b) $n$ for odd $n, n-1$ for even $n$.

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