Answer: 618

Small Hint:

Write out terms: $1000 - x,$ $2x - 1000,$ $2000 - 3x,$ $5x - 3000, \ldots$ — the coefficients are Fibonacci numbers

Big Hint:

Keeping the twelfth and thirteenth terms nonnegative pins $x$ between $\frac{55000}{89}$ and $\frac{89000}{144},$ an interval containing exactly one integer

Solution:

Computing terms, $a_3 = 1000 - x,$ $a_4 = 2x - 1000,$ $a_5 = 2000 - 3x,$ $a_6 = 5x - 3000,$ and in general $$\begin{aligned} a_{2k+1} &= 1000 F_{2k-1} - x F_{2k}, \\ a_{2k+2} &= x F_{2k+1} - 1000 F_{2k}, \end{aligned}$$ where $F_1 = F_2 = 1,$ $F_3 = 2, \ldots$ are the Fibonacci numbers. The sequence keeps going exactly as long as its terms stay nonnegative, so a long sequence requires $\frac{x}{1000}$ to be squeezed between the ratios $\frac{F_{2k}}{F_{2k+1}}$ and $\frac{F_{2k-1}}{F_{2k}}$ for larger and larger $k.$

For the first $13$ terms to be nonnegative we need $a_{12} = 89x - 55000 \ge 0$ and $a_{13} = 89000 - 144x \ge 0,$ i.e. $617.9\ldots \le x \le 618.05\ldots,$ so $x = 618.$ If $x \le 617$ the sequence turns negative by $a_{12},$ and if $x \ge 619$ it turns negative by $a_{13},$ so every other integer gives a shorter sequence.

Indeed $x = 618$ yields $1000, 618, 382, 236, 146,$ $90, 56, 34, 22, 12,$ $10, 2, 8, -6,$ a sequence of $14$ terms, the maximum possible. The answer is $618.$