Denote $a_n, n\in\{1,2\dots,39\}$ the terms of the sequence. For all $m\in\{1,2,\dots,19\}$, the sum of consecutive $2m$ terms is positive. $a_2+a_3+\dots+a_{38}+a_{39}>0$ and $S=a_1+a_2+a_3+\dots+a_{38}+a_{39}<0\Longrightarrow a_1<0$. $a_1+a_2+\dots+a_{37}+a_{38}>0$ and $S<0\Longrightarrow a_{39}<0$. For any $m\in\{1,2,\dots,18\}$, we can write: $S=(a_1+a_2+\dots+a_{2m})+a_{2m+1}+(a_{2m+2}+a_{2m+3}+\dots+a_{39})<0$; $a_1+a_2+\dots+a_{2m}>0$ (as sum of consecutive $2m$ terms); $a_{2m+2}+a_{2m+3}+\dots+a_{39}>0$ (as sum of consecutive $38-2m$ terms). Results: $a_{2m+1}<0, \forall m\in\{0,1,2,\dots,19\}$. $a_{2m}+a_{2m+1}>0$, hence $a_{2m}>0, \forall m\in\{1,2,\dots,19\}$. Conclusion: The sequence $(a_n)$ contains exactly $20$ negative terms ($a_1,a_3,a_5,\dots,a_{39}$), hence the product $P=a_1\cdot a_2\cdot ...\cdot a_{39}$ is positive.