Danube Mathematical Competition, 2014

* If $a_n + a_{2n-1} \ge 0$, then all the sums $a_i + a_{2n-1}$ with $i = \overline{n}, 2n-2$ as well as all the sums $a_i + a_{2n}$ with $i = \overline{n}, 2n-1$ are non-negative. In total, there are at least $(n-1)+n = 2n-1$ non-negative sums.

* If $a_n + a_{2n-1} < 0$, then $a_1 + \dots + a_{n-1} + a_{n+1} + \dots + a_{2n-2} + a_{2n} > 0$. (1)
We have $0 > a_n + a_{2n-1} \ge a_{n-1} + a_{2n-2} \ge \dots \ge a_2 + a_{n+1}$. It follows that $a_2 + a_3 + \dots + a_{n-1} + a_{n+1} + \dots + a_{2n-3} + a_{2n-2} < 0$. Combining this with (1) gives $a_1 + a_{2n} \ge 0$, hence all the sums $a_i + a_{2n}$ with $i = \overline{1, 2n-1}$ are non-negative.