ÖMO 1983

For every natural number $x$ , let $Q(x)$ be the sum and $P(x)$ the product of the (decimal) digits of $x$ . Show that for each $n \in \mathbb{N}$ there exist infinitely many values of $x$ such that:

$Q(Q(x))+P(Q(x))+Q(P(x))+P(P(x))=n$ .

Let $x_1,x_2,x_3$ be the roots of: $x^3-6x^2+ax+a=0$ . Find all real numbers $a$ for which $(x_1-1)^3+(x_2-1)^3+(x_3-1)^3=0$ . Also, for each such $a$ , determine the corresponding values of $x_1,x_2,$ and $x_3$ .

Let $P$ be a point in the plane of a triangle $ABC$ . Lines $AP,BP,CP$ respectively meet lines $BC,CA,AB$ at points $A',B',C'$ . Points $A'',B'',C''$ are symmetric to $A,B,C$ with respect to $A',B',C',$ respectively. Show that: $S_{A''B''C''}=3S_{ABC}+4S_{A'B'C'}$ .

The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_1=2, x_2=3,$ and

$x_{2m+1}=x_{2m}+x_{2m-1}$ for $m \ge 1;$ $x_{2m}=x_{2m-1}+2x_{2m-2}$ for $m \ge 2.$

Determine $x_n$ as a function of $n$ .

Given positive integers $a,b,$ find all positive integers $x,y$ satisfying the equation: $x^{a+b}+y=x^a y^b$ .

Planes $\pi _1$ and $\pi _2$ in Euclidean space $\mathbb{R} ^3$ partition $S=\mathbb{R} ^3 \setminus (\pi _1 \cup \pi _2)$ into several components. Show that for any cube in $\mathbb{R} ^3$ , at least one of the components of $S$ meets at least three faces of the cube.