MathDB

3

Part of 2015 Bundeswettbewerb Mathematik Germany

Problems(2)

Weird angle condition results into concurrence

Source: BWM Germany Round 1, 2015 - Problem 3

4/8/2015
Let MM be the midpoint of segment [AB][AB] in triangle △ABC\triangle ABC. Let XX and YY be points such that ∠BAX=∠ACM\angle{BAX}=\angle{ACM} and ∠BYA=∠MCB\angle{BYA}=\angle{MCB}. Both points, XX and YY, are on the same side as CC with respect to line ABAB. Show that the rays [AX[AX and [BY[BY intersect on line CMCM.
geometryBWMGermanyrayconcurrenceangleTriangle
Coloring 1,2,...,n with rules

Source: Bundeswettbewerb Mathematik 2015 - Round 2 - #3

9/4/2015
Each of the positive integers 1,2,…,n1,2,\dots,n is colored in one of the colors red, blue or yellow regarding the following rules: (1) A Number xx and the smallest number larger than xx colored in the same color as xx always have different parities. (2) If all colors are used in a coloring, then there is exactly one color, such that the smallest number in that color is even. Find the number of possible colorings.
Coloringcombinatoricscounting