MathDB

1

Part of 2011 Spain Mathematical Olympiad

Problems(2)

Coloring the edges between 67 vertices

Source: Spanish Mathematical Olympiad, National Stage, 2011

8/9/2011
Each pair of vertices of a regular 6767-gon is joined by a line segment. Suppose nn of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of nn for which, regardless of which nn segments were selected and how they were painted, there will always be a vertex of the polygon that belongs to seven segments of the same color.
combinatorics unsolvedcombinatorics
Congruent angles at a midpoint of a side in obtuse triangle

Source: Spanish Mathematical Olympiad, National Stage, 2011

8/9/2011
In triangle ABCABC, ∠B=2∠C\angle B=2\angle C and ∠A>90∘\angle A>90^\circ. Let DD be the point on the line ABAB such that CDCD is perpendicular to ACAC, and let MM be the midpoint of BCBC. Prove that ∠AMB=∠DMC\angle AMB=\angle DMC.
ratiogeometrytrapezoidtrigonometryperpendicular bisectorangle bisectorgeometry unsolved