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Problem 6 of Finals

Source: IX International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

September 22, 2018

floor functionstraight linesIntersection

Problem Statement

There are

a

straight lines in a plane, no two of which are parallel to each other and no three intersect in one point.
a) Prove that there exist a straight line for which each of the two Half-Planes defined by it contains at least

\lfloor \frac{(a-1)(a-2)}{10} \rfloor

intersection points.
b) Find all

a

for which the evaluation in a) is the best possible.

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