ÖMO 1988

If $a_1,...,a_{1988}$ are positive numbers whose arithmetic mean is $1988$ , show that:

$\sqrt[1988]{ \prod_{i,j=1}^{1988} \left( 1+\frac{a_i}{a_j} \right)} \ge 2^{1988}$

and determine when equality holds.

An equilateral triangle $A_1 A_2 A_3$ is divided into four smaller equilateral triangles by joining the midpoints $A_4,A_5,A_6$ of its sides. Let $A_7,...,A_{15}$ be the midpoints of the sides of these smaller triangles. The $15$ points $A_1,...,A_{15}$ are colored either green or blue. Show that with any such colouring there are always three mutually equidistant points $A_i,A_j,A_k$ having the same color.

Show that there is precisely one sequence $a_1,a_2,...$ of integers which satisfies $a_1=1, a_2>1,$ and $a_{n+1}^3+1=a_n a_{n+2}$ for $n \ge 1$ .

Let $a_{ij}$ be nonnegative integers such that $a_{ij}=0$ if and only if $i>j$ and that $\sum_{j=1}^{1988}a_{ij}=1988$ holds for all $i=1,...,1988$ . Find all real solutions of the system of equations:

$\sum_{j=1}^{1988} (1+a_{ij})x_j=i+1, 1 \le i \le 1988$ .

The bisectors of angles $B$ and $C$ of triangle $ABC$ intersect the opposite sides in points $B'$ and $C'$ respectively. Show that the line $B'C'$ intersects the incircle of the triangle.

Determine all monic polynomials $p(x)$ of fifth degree having real coefficients and the following property: Whenever $a$ is a (real or complex) root of $p(x)$ , then so are $\frac{1}{a}$ and $1-a$ .