MathDB

8

Part of 2005 Ukraine National Mathematical Olympiad

Problems(3)

construct a point

Source: Ukraine 2005 grade 9

7/26/2009
Let AB AB and CD CD be two disjoint chords of a circle. A point E E, distinct from A A and B B, is taken on the chord AB AB. Consider the arc AB AB not containing C C and D D. Using a ruler and a compass, construct a point F F on this arc such that \frac{PE}{EQ}\equal{}\frac{1}{2}, where P P and Q Q are the intersection points of AB AB with the segments FC FC and FD FD, respectively.
geometry unsolvedgeometry
points

Source: Ukraine 2005 grade 10

7/28/2009
On the plane are marked 2005 2005 points, no three of which are collinear. A line is drawn through any two of the points. Show that the points can be painted in two colors so that for any two points of the same color the number of the drawn lines separating them is even. (Two points are separated by a line if they lie in different open half-planes determined by the line).
combinatorics proposedcombinatorics
colored points

Source: Ukraine 2005 grade 11

7/28/2009
In space are marked 2005 2005 points, no four of which are in the same plane. A plane is drawn through any three points. Show that the points can be painted in two colors so that for any two points of the same color the number of the drawn planes separating them is odd. (Two points are separated by a plane if they lie in different open half-spaces determined by the plane).
combinatorics proposedcombinatorics