ZA-SAMO 2001

$ABCD$ is a convex quadrilateral with perimeter $p$ . Prove that $$\dfrac{1}{2}p < AC + BD < p.$$ (A polygon is convex if all of its interior angles are less than $180^\circ$ .)

Find all triples $(x,y,z)$ of real numbers that satisfy $$\begin{aligned} & x\left(1 - y^2\right)\left(1 - z^2\right) + y\left(1 - z^2\right)\left(1 - x^2\right) + z\left(1 - x^2\right)\left(1 - y^2\right) & = 4xyz & = 4(x + y + z). \end{aligned}$$

For a certain real number $x$ , the differences between $x^{1919}$ , $x^{1960}$ and $x^{2001}$ are all integers. Prove that $x$ is an integer.

$n$ red and $n$ blue points on a plane are given so that no three of the $2n$ points are collinear. Prove that it is always possible to split up the points into $n$ pairs, with one red and one blue point in each pair, so that no two of the $n$ line segments which connect the two members of a pair intersect.

Starting from a given cyclic quadrilateral $\mathcal{Q}_0$ , a sequence of quadrilaterals is constructed so that $\mathcal{Q}_{k + 1}$ is the circumscribed quadrilateral of $\mathcal{Q}_k$ for $k = 0,1,\dots$ . The sequence terminates when a quadrilateral is reached that is not cyclic. (The circumscribed quadrilateral of a cylic quadrilateral $ABCD$ has sides that are tangent to the circumcircle of $ABCD$ at $A$ , $B$ , $C$ and $D$ .) Prove that the sequence always terminates, except when $\mathcal{Q}_0$ is a square.

The unknown real numbers $x_1,x_2,\dots,x_n$ satisfy $x_1 < x_2 < \cdots < x_n,$ where $n \geq 3$ . The numbers $s$ , $t$ and $d_1,d_2,\dots,d_{n - 2}$ are given, such that $$\begin{aligned} s & = \sum\limits_{i = 1}^nx_i, t & = \sum\limits_{i = 1}^nx_i^2, d_i & = x_{i + 2} - x_i,\ \ i = 1,2,\dots,n - 2. \end{aligned}$$ For which $n$ is this information always sufficient to determine $x_1,x_2,\dots,x_n$ uniquely?