### 11^{th} 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 *OP*^{2} where
*A* and *B* are the intersections of *OX* and *OY*
with *d*.
### Problem 2

Prove that the polynomial *x*^{4} - 1993 *x*^{3}
+ (1993 + *m*) *x*^{2} - 11 *x* + *m* has at
most one integer root.
### Problem 3

For all the permutations (*a*_{1}, *a*_{2},...
, *a*_{n}) of the set {1, 2,..., *n*} where *n* is
a fixed positive integer, find the maximum value of the sum
|*a*_{1} - *a*_{2}| + |*a*_{2} -
*a*_{3}| + ... + |*a*_{n-1} - *a*_{n}|
### 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.