Express in terms of $\sigma_{1}, \ldots, \sigma_{n}$:
a) $\sum_{i=1}^{n} \frac{1}{x_{i}}$
b) $\sum_{i=1}^{n} x_{i}^{2}$
c) $\sum_{i \neq j} x_{i} x_{j}^{2}$.
a) $\sigma_{n-1} / \sigma_{n}$
b) $\sigma_{1}^{2}-2 \sigma_{2}$

c) On $\sigma_{1} \sigma_{2}=\sum_{i} \sum_{j<k} x_{i} x_{j} x_{k}$.

If $a<b<c$, the term $x_{a} x_{b} x_{c}$ appears 3 times because $(j, k)$ can be $(a, b),(b, c)$ or $(a, c)$.

If $a \neq b$, the term $x_{a} x_{b}^{2}$ appears once because $\{j, k\}=\{a, b\}$ and $i=b$.

We deduce that $\sum_{i \neq j} x_{i} x_{j}^{2}=\sigma_{1} \sigma_{2}-3 \sigma_{3}$.