There are positive real numbers $a_1, \dots ,a_{2013}$, that satisfy: equations $x_{k-1}-2x_{k}+x_{k+1}+a_{k}x_{k}=0(1 \le k \le 2013)$ have a solution $(x_0,x_1,\dots ,x_{2014})(x_0=x_{2014}=0)$ which is not all zeros.
Prove that $a_1+\dots +a_{2013} \ge \frac{2}{1007}$
It likely needs a smart algebraic transform. I have asked on AOPS, but no one responded.