FI-MO 2008

Foxes, wolves and bears arranged a big rabbit hunt. There were $45$ hunters catching $2008$ rabbits. Every fox caught $59$ rabbits, every wolf $41$ rabbits and every bear $40$ rabbits. How many foxes, wolves and bears were there in the hunting company?

The incentre of the triangle $ABC$ is $I.$ The lines $AI, BI$ and $CI$ meet the circumcircle of the triangle $ABC$ also at points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.

Solve the diophantine equation $$x^{2008}- y^{2008} = 2^{2009}.$$

Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches. What is the largest possible number of matches?

The closed line segment $I$ is covered by finitely many closed line segments. Show that one can choose a subfamily $S$ of the family of line segments having the properties: (1) the chosen line segments are disjoint, (2) the sum of the lengths of the line segments of S is more than half of the length of $I.$ Show that the claim does not hold any more if the line segment $I$ is replaced by a circle and other occurences of the compound word ''line segment" by the word ''circular arc".