MathDB

3

Part of 2006 All-Russian Olympiad

Problems(2)

A game with 2006 points and chords on a circle

Source: All-Russian Olympiad 2006 finals, problem 10.3 = 9.3

5/7/2006
Given a circle and 20062006 points lying on this circle. Albatross colors these 20062006 points in 1717 colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common endpoint). Hereby, Frankinfueter intends to draw as many chords as possible, while Albatross is trying to hinder him as much as he can. What is the maximal number of chords Frankinfueter will always be able to draw?
inductionalgebra proposedalgebracombinatorics
Directing segments: who wins?

Source: All-Russian Olympiad 2006 finals, problem 11.3

5/6/2006
On a 49×6949\times 69 rectangle formed by a grid of lattice squares, all 50⋅7050\cdot 70 lattice points are colored blue. Two persons play the following game: In each step, a player colors two blue points red, and draws a segment between these two points. (Different segments can intersect in their interior.) Segments are drawn this way until all formerly blue points are colored red. At this moment, the first player directs all segments drawn - i. e., he takes every segment AB, and replaces it either by the vector AB→\overrightarrow{AB}, or by the vector BA→\overrightarrow{BA}. If the first player succeeds to direct all the segments drawn in such a way that the sum of the resulting vectors is 0→\overrightarrow{0}, then he wins; else, the second player wins. Which player has a winning strategy?
geometryrectanglevectortrigonometrycombinatorics proposedcombinatorics