Bulgarian Mathematical Competitions

Solution:
We first prove that at least one of the numbers $a_{1}, a_{2}, \ldots, a_{2005}$ is not positive. To do this we assume the contrary and choose $i$ such that
$$\frac{b_{i}}{a_{i}}=M=\max _{1 \leq j \leq 2005}\left(\frac{b_{j}}{a_{j}}\right)$$
Then we can find $\varepsilon>0$ such that
$$\left(a_{i} x-b_{i}\right)^{2}<\sum_{j=1, j \neq i}^{2005}\left(a_{j} x-b_{j}\right)$$
for every $x \in(M, M+\varepsilon)$, a contradiction.

On the other hand, it is easy to see that if $a_{1}=a_{2}=\cdots=a_{2004}=-a_{2005}=1$ and $b_{1}=b_{2}=\cdots=b_{2004}=b_{2005} \geq \frac{1001^{2}}{2}$ the given inequality is satisfied. Therefore the answer is 4009.