14th BALKAN MATHEMATICAL OLYMPIAD
Kalampaka, Greece, April 29, 1997
Problem 1
Let O be an interior point of a convex quadrilateral ABCD satisfaying
OA2 + OB2 + OC2 +
OD2 = 2 S(ABCD)
where S(ABCD) denotes the area of ABCD. Show that
ABCD is a square with center O.
Problem 2
Let A = {A1, A2, ... ,
Ak} (k > 1), be a collection of subsets of
an
n-set S such that for any x, y Î S there is Ai
Î A such that x Î Ai and
y Î A \ Ai or
x Î A \ Ai
and y Î Ai. Show that
k ³
[log2 n].
Problem 3
Three circles G, C1 and
C2 are given in the plane. C1
and C2 touch G internally at
points B and C, respectively.
Moreover, C1 and C2 touch each other
externally at a point D. Let A
be one of the points in which the common tangent of C1
and C2 intersects G.
Denote by M the second point of intersection of the line AB
and the circle C1 and by N the second
point of intersection of the line AC and the circle C2. Further denote by K and L
the second points of intersection of the line BC with
C1 and C2, respectively.
Show that the lines AD, MK and NL are concurrent.
Problem 4
Find all functions f : R ® R such that
f (xf(x) + f(y)) = (f (x))2 + y
holds for all x and y.