BALKAN 1994

An acute angle $XAY$ and a point $P$ inside the angle are given. Construct (using a ruler and a compass) a line that passes through $P$ and intersects the rays $AX$ and $AY$ at $B$ and $C$ such that the area of the triangle $ABC$ equals $AP^2$ .

Greece

Let $n$ be an integer. Prove that the polynomial $f(x)$ has at most one zero, where $$f(x) = x^4 - 1994 x^3 + (1993+n)x^2 - 11x + n .$$ Greece

Let $a_1,a_2,\ldots,a_n$ be a permutation of the numbers $1,2,\ldots,n$ , with $n\geq 2$ . Determine the largest possible value of the sum $$S(n)=|a_2-a_1|+ |a_3-a_2| + \cdots + |a_n-a_{n-1}| .$$ Romania

Find the smallest number $n \geq 5$ for which there can exist a set of $n$ people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances.

Bulgaria