RagvaloD wrote: There are $2008$ trinomials $x^2-a_kx+b_k$ where $a_k$ and $b_k$ are all different numbers from set $(1,2,...,4016)$. These trinomials has not common real roots. We mark all real roots on the $Ox$-axis. Prove, that distance between some two marked points is $\leq \frac{1}{250}$ All these trinomial have real root?

There are such condition, so I think , that answer is no.