5

Part of 1999 May Olympiad

Problems(2)

12 vertices fo regular 12-gon

Source: V May Olympiad (Olimpiada de Mayo) 1999 L2 P5

12

points that are vertices of a regular polygon with

12

sides. Rafael must draw segments that have their two ends at two of the points drawn. He is allowed to have each point be an endpoint of more than one segment and for the segments to intersect, but he is prohibited from drawing three segments that are the three sides of a triangle in which each vertex is one of the

12

starting points. Find the maximum number of segments Rafael can draw and justify why he cannot draw a greater number of segments.

combinatoricscombinatorial geometry

Ana, Beatriz, Carlos, Diego and Emilia play a chess tournament

Source: V May Olympiad (Olimpiada de Mayo) 1999 L1 P5

Ana, Beatriz, Carlos, Diego and Emilia play a chess tournament. Each player faces each of the other four only once. Each player gets

2

points if he wins the match,

1

point if he draws and

0

point if he loses. At the end of the tournament, it turns out that the scores of the

5

players are all different. Find the maximum number of ties there could be in the tournament and justify why there could not be a higher number of ties.

combinatoricsalgebra