MathDB
Space quadrilateral

Source: 2003 China Second Round Olympiad

August 30, 2014
geometry unsolvedgeometry

Problem Statement

Let a space figure consist of nn vertices and ll lines connecting these vertices, with n=q2+q+1n=q^2+q+1, l≥q2(q+1)2+1l\ge q^2(q+1)^2+1, q≥2q\ge2, q∈Nq\in\mathbb{N}. Suppose the figure satisfies the following conditions: every four vertices are non-coplaner, every vertex is connected by at least one line, and there is a vertex connected by at least p+2p+2 lines. Prove that there exists a space quadrilateral in the figure, i.e. a quadrilateral with four vertices A,B,C,DA, B, C, D and four lines AB,BC,CD,DA AB, BC, CD, DA in the figure.
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