  Let $\bar{X}_{n}=(x_{1},\ldots,x_{n})$ and $\sigma_{i}(\bar{X}_{n})=\sum
x_{k_{1}}\ldots x_{k_{i}}$ be $i$-th elementary symmetric polynomial. In this
note we prove that there are infinitely many triples of integers $a, b, c$ such
that for each $1\leq i\leq n$ the system of Diophantine equations
\begin{equation*}
  \sigma_{i}(\bar{X}_{2n})=a, \quad \sigma_{2n-i}(\bar{X}_{2n})=b, \quad
\sigma_{2n}(\bar{X}_{2n})=c \end{equation*} has infinitely many rational
solutions. This result extend the recent results of Zhang and Cai, and the
author. Moreover, we also consider some Diophantine systems involving sums of
powers. In particular, we prove that for each $k$ there are at least $k$
$n$-tuples of integers with the same sum of $i$-th powers for $i=1,2,3$.
Similar result is proved for $i=1,2,4$ and $i=-1,1,2$.
