OIM 1991

Each vertex of a cube is assigned an 1 or a -1, and each face is assigned the product of the numbers assigned to its vertices. Determine the possible values the sum of these 14 numbers can attain.

A square is divided in four parts by two perpendicular lines, in such a way that three of these parts have areas equal to 1. Show that the square has area equal to 4.

Let $f: \ [0,\ 1] \rightarrow \mathbb{R}$ be an increasing function satisfying the following conditions:

a) $f(0)=0$ ;

b) $f\left(\frac{x}{3}\right)=\frac{f(x)}{2}$ ;

c) $f(1-x)=1-f(x)$ .

Determine $f\left(\frac{18}{1991}\right)$ .

Find a positive integer $n$ with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of $n$ .

Let $P(x, y)=2x^{2}-6xy+5y^{2}$ . Let us say an integer number $a$ is a value of $P$ if there exist integer numbers $b$ , $c$ such that $P(b, c)=a$ .

a) Find all values of $P$ lying between 1 and 100.

b) Show that if $r$ and $s$ are values of $P$ , then so is $rs$ .

Let $M$ , $N$ and $P$ be three non-collinear points. Construct using straight edge and compass a triangle for which $M$ and $N$ are the midpoints of two of its sides, and $P$ is its orthocenter.