AUSTRIAN-POLISH 1985

Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$ .

Suppose that $n\ge 8$ persons $P_1,P_2,\dots,P_n$ meet at a party. Assume that $P_k$ knows $k+3$ persons for $k=1,2,\dots,n-6$ . Further assume that each of $P_{n-5},P_{n-4},P_{n-3}$ knows $n-2$ persons, and each of $P_{n-2},P_{n-1},P_n$ knows $n-1$ persons. Find all integers $n\ge 8$ for which this is possible.

(It is understood that "to know" is a symmetric nonreflexive relation: if $P_i$ knows $P_j$ then $P_j$ knows $P_i$ ; to say that $P_i$ knows $p$ persons means: knows $p$ persons other than herself/himself.)

In a convex quadrilateral of area $1$ , the sum of the lengths of all sides and diagonals is not less than $4+\sqrt 8$ . Prove this.

Solve the system of equations: $\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$

We are given a certain number of identical sets of weights; each set consists of four different weights expressed by natural numbers (of weight units). Using these weights we are able to weigh out every integer mass up to $1985$ (inclusive). How many ways are there to compose such a set of weight sets given that the joint mass of all weights is the least possible?

Let $P$ be a point inside a tetrahedron $ABCD$ and let $S_A,S_B,S_C,S_D$ be the centroids (i.e. centers of gravity) of the tetrahedra $PBCD,PCDA,PDAB,PABC$ . Show that the volume of the tetrahedron $S_AS_BS_CS_D$ equals $1/64$ the volume of $ABCD$ .

Find an upper bound for the ratio $$\frac{x_1x_2+2x_2x_3+x_3x_4}{x_1^2+x_2^2+x_3^2+x_4^2}$$ over all quadruples of real numbers $(x_1,x_2,x_3,x_4)\neq (0,0,0,0)$ .

Note. The smaller the bound, the better the solution.

A convex $n$ -gon $A_0A_1\dots A_{n-1}$ has been partitioned into $n-2$ triangles by certain diagonals not intersecting inside the $n$ -gon. Prove that these triangles can be labeled $\triangle_1,\triangle_2,\dots,\triangle_{n-2}$ in such a way that $A_i$ is a vertex of $\triangle_i$ , for $i=1,2,\dots,n-2$ . Find the number of all such labellings.

We are given a convex polygon. Show that one can find a point $Q$ inside the polygon and three vertices $A_1,A_2,A_3$ (not necessarily consecutive) such that each ray $A_iQ$ ( $i=1,2,3$ ) makes acute angles with the two sides emanating from $A_i$ .