Solve the system of equations

$$
\left\{\begin{array}{l}
x+y+z=0 \\
x y z=-1 \\
x^{2}+y^{2}+z^{2}=4
\end{array}\right.
$$
We denote $\sigma_{k}$ as the elementary symmetric functions in $x, y, z$. We have $\sigma_{1}=0, \sigma_{3}=-1$ and $\sigma_{1}^{2}-2 \sigma_{2}=4$, so $\sigma_{2}=-2$. From this, we deduce that $x, y, z$ are the roots of $X^{3}-2 X+1$. This polynomial has an obvious root of 1. By performing Euclidean division by $X-1$, we get $X^{3}-2 X+1=(X-1)\left(X^{2}+X-1\right)$, so $x, y, z$ form, in some order, the list $1, \frac{-1+\sqrt{5}}{2}, \frac{-1-\sqrt{5}}{2}$.

## 2 Thursday morning: Matthieu Lequesne

## A square is always non-negative

Theorem 6. A square is always non-negative: $\forall x \in \mathbb{R}, \quad x^{2} \geq 0$. More generally:

$$
\forall n \in \mathbb{N}, \forall\left(x_{1} \ldots x_{n}\right) \in \mathbb{R}^{n}, \quad \sum_{k=1}^{n} x_{i}^{2} \geq 0
$$

We are therefore looking to reduce the expression to a form like $\sum_{k=1}^{n} x_{i}^{2} \geq 0$. Many equalities are equivalent to an expression of this type.