BG-MO 1993

Find all functions $f$ , defined and having values in the set of integer numbers, for which the following conditions are satisfied: (a) $f(1) = 1$ ; (b) for every two whole (integer) numbers $m$ and $n$ , the following equality is satisfied: $$f(m+n)·(f(m)-f(n)) = f(m-n)·(f(m)+ f(n))$$

Let $M$ be an interior point of the triangle $ABC$ such that $AMC = 90^\circ$ , $AMB = 150^\circ$ , and $BMC = 120^\circ$ . The circumcenters of the triangles $AMC$ , $AMB$ , and $BMC$ are $P$ , $Q$ , and $R$ respectively. Prove that the area of $\Delta PQR$ is greater than or equal to the area of $\Delta ABC$ .

it is given a polyhedral constructed from two regular pyramids with bases heptagons (a polygon with $7$ vertices) with common base $A_1A_2A_3A_4A_5A_6A_7$ and vertices respectively the points $B$ and $C$ . The edges $BA_i , CA_i$ $(i = 1,...,7$ ), diagonals of the common base are painted in blue or red. Prove that there exists three vertices of the polyhedral given which forms a triangle with all sizes in the same color.

Find all natural numbers $n > 1$ for which there exists such natural numbers $a_1,a_2,...,a_n$ for which the numbers $\{a_i +a_j | 1 \le i \le j \le n \}$ form a full system modulo $\frac{n(n+1)}{2}$ .

Let $Oxy$ be a fixed rectangular coordinate system in the plane. Each ordered pair of points $A_1, A_2$ from the same plane which are different from O and have coordinates $x_1, y_1$ and $x_2, y_2$ respectively is associated with real number $f(A_1,A_2)$ in such a way that the following conditions are satisfied:

(a) If $OA_1 = OB_1$ , $OA_2 = OB_2$ and $A_1A_2 = B_1B_2$ then $f(A_1,A_2) = f(B_1,B_2)$ .

(b) There exists a polynomial of second degree $F(u,v,w,z)$ such that $f(A_1,A_2)=F(x_1,y_1,x_2,y_2)$ .

(c) There exists such a number $\phi \in (0,\pi)$ that for every two points $A_1, A_2$ for which $\angle A_1OA_2 = \phi$ is satisfied $f(A_1,A_2) = 0$ .

(d) If the points $A_1, A_2$ are such that the triangle $OA_1A_2$ is equilateral with side $1$ then $f(A_1,A_2) = \frac12$ .

Prove that $f(A_1,A_2) = \overrightarrow{OA_1} \cdot \overrightarrow{OA_2}$ for each ordered pair of points $A_1, A_2$ .

Find all natural numbers $n$ for which there exists set $S$ consisting of $n$ points in the plane, satisfying the condition: For each point $A \in S$ there exist at least three points say $X, Y, Z$ from $S$ such that the segments $AX, AY$ and $AZ$ have length $1$ (it means that $AX = AY = AZ = 1$ ).