AUSTRIAN-POLISH 1989

Show that $(\sum_{i=1}^{n}x_iy_iz_i)^2 \le (\sum_{i=1}^{n}x_i^3) (\sum_{i=1}^{n}y_i^3) (\sum_{i=1}^{n}z_i^3)$ for any positive reals $x_i, y_i, z_i$ .

Each point of the plane is colored by one of the two colors. Show that there exists an equilateral triangle with monochromatic vertices.

Find all natural numbers $N$ (in decimal system) with the following properties: (i) $N =\overline{aabb}$ , where $\overline{aab}$ and $\overline{abb}$ are primes, (ii) $N = P_1P_2P_3$ , where $P_k (k = 1,2,3)$ is a prime consisting of $k$ (decimal) digits.

Let $P$ be a convex polygon in the plane. Show that there exists a circle containing the entire polygon $P$ and having at least three adjacent vertices of $P$ on its boundary.

Let $A$ be a vertex of a cube $\omega$ circumscribed about a sphere $k$ of radius $1$ . We consider lines $g$ through $A$ containing at least one point of $k$ . Let $P$ be the intersection point of $g$ and $k$ closer to $A$ , and $Q$ be the second intersection point of $g$ and $\omega$ . Determine the maximum value of $AP\cdot AQ$ and characterize the lines $g$ yielding the maximum.

A sequence $(a_n)_{n \in N}$ of squares of nonzero integers is such that for each $n$ the difference $a_{n+1} - a_n$ is a prime or the square of a prime. Show that all such sequences are finite and determine the longest sequence.

Functions $f_0, f_1,f_2,...$ are recursively defined by $f_0(x) = x$ and $f_{2k+1} (x) = 3^{f_{2k}(x)}$ and $f_{2k+2} = 2^{f_{2k+1}(x)}$ , $k = 0,1,2,...$ for all $x \in R$ . Find the greater one of the numbers $f_{10}(1)$ and $f_9(2)$ .

$ABC$ is an acute-angled triangle and $P$ a point inside or on the boundary. The feet of the perpendiculars from $P$ to $BC, CA, AB$ are $A', B', C'$ respectively. Show that if $ABC$ is equilateral, then $\frac{AC'+BA'+CB'}{PA'+PB'+PC'}$ is the same for all positions of $P$ , but that for any other triangle it is not.

Find the smallest odd natural number $N$ such that $N^2$ is the sum of an odd number (greater than $1$ ) of squares of adjacent positive integers.