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possible to draw straight line through midpoints of at least 506 pieces of wire

Source: Thailand Mathematical Olympiad 2015 p3

August 16, 2020
combinatoricscombinatorial geometry

Problem Statement

Let P={(x,y)∣x,y∈{0,1,2,...,2015}}P = \{(x, y) | x, y \in \{0, 1, 2,... , 2015\}\} be a set of points on the plane. Straight wires of unit length are placed to connect points in PP so that each piece of wire connects exactly two points in PP, and each point in PP is an endpoint of exactly one wire. Prove that no matter how the wires are placed, it is always possible to draw a straight line parallel to either the horizontal or vertical axis passing through midpoints of at least 506506 pieces of wire.
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