Problem 4
A set X of positive integers is called nice if for each pair a, b belong to X
exactly one of the numbers a+b and |a-b| belongs to X (the numbers a and b may
be different). Determine the number of nice sets containing the number 2008.
Problem 5
A group of people is called good if its members can be distributed to several
rooms so that nobody is acquainted with any person in the same room but it is
possible to choose a person from each room so that all the chosen persons are
acquainted with each other.
A group is called perfect if it is good and every set of its members is also
good.
A perfect group planned a party. However one of its members, Alice, brought
her acquaintance Bob, who was not originally expected, and introduced him to
all her other acquaintances. Prove that the new group is also perfect.
Problem 6
A convex hexagon is given. Let S be the sum of the lengths of the three
segments connecting the midpoints of its opposite sides. Prove that there is
a point in the hexagon such that the sum of its distances to the lines
containing the sides of the hexagon does not exceed S.
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