ÖMO 1987

The sides $a,b$ and the bisector of the included angle $\gamma$ of a triangle are given. Determine necessary and sufficient conditions for such triangles to be constructible and show how to reconstruct the triangle.

Find the number of all sequences $(x_1,...,x_n)$ of letters $a,b,c$ that satisfy $x_1=x_n=a$ and $x_i \not= x_{i+1}$ for $1 \le i \le n-1.$

Let $x_1,...,x_n$ be positive real numbers. Prove that:

$\sum_{k=1}^{n}x_k+\sqrt{ \sum_{k=1}^{n}x_k^2} \le \frac{n+\sqrt{n}}{n^2} \left( \sum_{k=1}^{n} \frac{1}{x_k} \right) \left( \sum_{k=1}^{n} x_k^2 \right).$

Find all triples $(x,y,z)$ of natural numbers satisfying $2xz=y^2$ and $x+z=1987$ .

Let $P$ be a point in the interior of a convex $n$ -gon $A_1 A_2 ... A_n$ $(n \ge 3)$ . Show that among the angles $\beta _{ij}=\angle A_i P A_j,1 \le i \le n$ , there are at least $n-1$ angles satisfying $90^{\circ} \le \beta_{ij} \le 180^{\circ}$ .

Determine all polynomials $P_n(x)=x^n+a_1 x^{n-1}+...+a_{n-1} x+a_n$ with integer coefficients whose $n$ zeros are precisely the numbers $a_1,...,a_n$ (counted with their respective multiplicities).