11th BALKAN MATHEMATICAL OLYMPIAD
NOVI SAD, YUGOSLAVIA, 1994
Problem 1
Consider the angle XOY and a point P inside this angle.
Construct the line d passing through P using ruler and
compasses such that the area of OAB is OP2 where
A and B are the intersections of OX and OY
with d.
Problem 2
Prove that the polynomial x4 - 1993 x3
+ (1993 + m) x2 - 11 x + m has at
most one integer root.
Problem 3
For all the permutations (a1, a2,...
, an) of the set {1, 2,..., n} where n is
a fixed positive integer, find the maximum value of the sum
|a1 - a2| + |a2 -
a3| + ... + |an-1 - an|
Problem 4
Find the least integer n > 4 such that there exists a set of
n persons with the following properties:
a) any two persons that do not know each other have no common
acquaintances
b) any two persons that do not know each other have exactly two
common friends.
It is assumed that if A has B as his friend then also B has A as his friend.