4
Part of 2007 Bundeswettbewerb Mathematik
Problems(2)
Solve this equation.
Source:
12/31/2006
Let be a positive integer.
How many non-negative integer solutions x does the equation
have?
—> [url=http://en.wikipedia.org/wiki/Floor_function]Floor Function.
floor functionfunctionalgebra
Dissected into 54 congruent equilateral triangles
Source: Germany Bundeswettbewerb Mathematik 2007, Round 2, Problem 4
7/18/2008
A regular hexagon, as shown in the attachment, is dissected into 54 congruent equilateral triangles by parallels to its sides. Within the figure we yield exactly 37 points which are vertices of at least one of those triangles. Those points are enumerated in an arbitrary way. A triangle is called clocky if running in a clockwise direction from the vertex with the smallest assigned number, we pass a medium number and finally reach the vertex with the highest number. Prove that at least 19 out of 54 triangles are clocky.
combinatorics unsolvedcombinatorics