Consider the following system of equations:

$$ \sum_{i=1}^{2n}a_i=0 $$ $$ \sum_{i=1}^{2n}\frac{1}{a_i}=0 $$ Where for each $i$ $a_i$ is an odd integer and the $a_i$ are not necessarly distinct. A solution $(a_1,\dots,a_{2n})$ is trivial if (after some permutation of the coefficients) for each $i$ we have $$a_i=-a_{n+i}$$. I know that if $n>2$ there exist non trivial solutions. My questions are:

- What is the minimum number of variables for which there exist non trivial solutions ?
- Can you exhibit a minimal solution or at least a solution you think could be minimal ?