Problem 1.
Let ABC be a triangle with ∠BAC≠90°. Let O be the circumcenter of
the triangle ABC and let Γ be the circumcircle of the triangle BOC.
Suppose that Γ inter sects the line segment AB at P different from B,
and the line segment AC at Q different from C. Let ON be a diameter of
the circle Γ. Prove that the quadrilateral APNQ is a parallelogram.
Problem 2.
For a positive integer k, call an integer a pure k-th power if it can
be represented as m^k for some integer m. Show that for every positive
integer n there exist n distinct positive integers such that their sum
is a pure 2009-th power, and their product is a pure 2010-th power.
Problem 3.
Let n be a positive integer. n people take part in a certain party.
For any pair of the participants, either the two are acquainted with
each other or they are not. What is the maximum possible number of the
pairs for which the two are not acquainted but have a common acquaintance
among the participants？
Problem 4.
Let ABC be an acute triangle satisfying the condition AB > BC and AC > BC.
Denote by O and H the circumcenter and the orthocenter, respectively,
of the triangle ABC. Suppose that the circumcircle of the triangle AHC
intersects the line AB at M different from A, and that the circumcircle of
the triangle AHB intersects the line AC at N different from A. Prove that
the circumcenter of the triangle MNH lies on the line OH.
Problem 5.
Find all functions f from the set R of real numbers into R which satisfy
for all x,y,z belonging to R the identity
f(f(x) + f(y) + f(z)) = f(f(x) - f(y)) + f(2xy + f(z)) + 2f(xz-yz).