MathDB

4

Part of 2011 South East Mathematical Olympiad

Problems(2)

The same angles

Source: China South East Mathematical Olympiad2011

8/18/2011
Let OO be the circumcenter of triangle ABCABC , a line passes through OO intersects sides AB,ACAB,AC at points M,NM,N , EE is the midpoint of MCMC , FF is the midpoint of NBNB , prove that : ∠FOE=∠BAC\angle FOE= \angle BAC
geometrycircumcircleprojective geometrygeometry proposed
The 12 points on the clock

Source: China South East Mathematical Olympiad2011

8/18/2011
12 points are located on a clock with the sme distance , numbers 1,2,3,...121,2,3 , ... 12 are marked on each point in clockwise order . Use 4 kinds of colors (red,yellow,blue,green) to colour the the points , each kind of color has 3 points . N ow , use these 12 points as the vertex of convex quadrilateral to construct nn convex quadrilaterals . They satisfies the following conditions: (1). the colours of vertex of every convex quadrilateral are different from each other . (2). for every 3 quadrilaterals among them , there exists a colour such that : the numbers on the 3 points painted into this colour are different from each other . Find the maximum nn .
combinatorics proposedcombinatorics