ILL 1989 Problem 26

Let $b_1, b_2, \ldots, b_{1989}$ be positive real numbers such that the equations

$$x_{r-1} - 2x_r + x_{r+1} + b_rx_r = 0 (1 \leq r \leq 1989)$$

have a solution with $x_0 = x_{1989} = 0$ but not all of $x_1, \ldots, x_{1989}$ are equal to zero. Prove that

$$\sum^{1989}_{k=1} b_k \geq \frac{2}{995}.$$