In the plane, $n$ lines have been drawn in such a way that no two lines are parallel and no three lines are concurrent. How many regions are delimited by these lines?
When there are no lines, there is 1 region. We denote $u_{n}$ as the number of regions delimited when there are $n$ lines in general position. Consider a configuration of $n$ lines in general position. When adding a line, it intersects each other line at 1 point. We number the intersection points by increasing abscissa. Before the first intersection point, the line cuts one region into 2 new ones. Between each intersection point, the line cuts another region into 2. After the last intersection point, the line cuts another region into 2. This results in $n+1$ regions. Therefore, $u_{n+1}=u_{n}+n+1$ for any natural number $n$. Thus:

$$
\begin{aligned}
u_{n}-u_{0} & =\left(u_{n}-u_{n-1}\right)+\left(u_{n-1}-u_{n-2}\right)+\ldots+\left(u_{2}-u_{1}\right)+u_{1}-u_{0} \\
& =n+(n-1)+(n-2)+\ldots+2+1 \\
& =\frac{n(n+1)}{2}
\end{aligned}
$$

We obtain for any integer $n$ that $u_{n}=\frac{n(n+1)}{2}+1$.

## 2 Wednesday, 25 morning: Vincent Jugé

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