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Part of 2008 Spain Mathematical Olympiad

Problems(2)

Colouring every point in the plane one of 7 colours

Source:

Every point in the plane is coloured one of seven distinct colours. Is there an inscribed trapezoid whose vertices are all of the same colour?

geometrytrapezoidrectanglecombinatorics unsolvedcombinatorics

Maximum number of regions after p-secting sides of triangle

Source:

p\ge 3

be a prime number. Each side of a triangle is divided into

p

equal parts, and we draw a line from each division point to the opposite vertex. Find the maximum number of regions, every two of them disjoint, that are formed inside the triangle.

modular arithmeticcombinatorics unsolvedcombinatorics