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Many roots

Source: St Petersburg Olympiad 2008, Grade 11, P3

August 30, 2017
algebra

Problem Statement

There are 20082008 trinomials x2−akx+bkx^2-a_kx+b_k where aka_k and bkb_k are all different numbers from set (1,2,...,4016)(1,2,...,4016). These trinomials has not common real roots. We mark all real roots on the OxOx-axis. Prove, that distance between some two marked points is ≤1250\leq \frac{1}{250}
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