VMO 1993 — Division A

Let $f:[\sqrt{1995},\infty)\to\mathbb R$ be defined by $f(x)=x(1993+\sqrt{1995-x^2})$. Determine its maximum and minimum values.

Let $ABCD$ be a quadrilateral such that $AB$ is not parallel to $CD$ and $BC$ is not parallel to $AD$. Variable points $P,Q,R,S$ lie respectively on $AB,BC,CD,DA$, with $PQRS$ a parallelogram. Determine the locus of its center.

Determine the function $f(n)$ defined on the positive integers, with positive integer values, satisfying $f(f(n))=1993n^{1945}$ for every positive integer $n$.

Let $ABCD$ be a tetrahedron whose vertices lie on a fixed sphere $S$. Determine all possible positions of the tetrahedron so that the expression $AB^2+AC^2+AD^2-BC^2-BD^2-CD^2$ [the printed question ends here].

There are $1993$ points arranged on a circle. Initially each is assigned either $+1$ or $-1$, arbitrarily. At steps $n=1,2,3,\ldots$, the labels are changed: at step $n$ a point is assigned $+1$ if its two neighboring points have equal labels at step $n-1$. Prove that at some step $n$ all labels are $1$.

Sequences $a_0,a_1,a_2,\ldots$ and $b_0,b_1,b_2,\ldots$ satisfy $a_0=2$, $b_0=1$,
$$a_{n+1}=\frac{2a_nb_n}{a_n+b_n},\qquad b_{n+1}=\sqrt{a_{n+1}b_n}.$$
Show that the two sequences converge to the same limit and determine that limit.