VMO 1988 — Division A

There are 1988 chickens housed in 994 cages, two in each cage. After each day, their positions are changed so that no two chickens that have previously shared a cage share one again. For how many days can this be done?

Let $P(x)=a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$, where $n\ge3$. Suppose that the polynomial has $n$ real roots and that $a_0=1$, $a_1=-n$, $a_2=(n^2-n)/2$. Determine the coefficients $a_i$ for $i=3,4,\ldots,n$.

The plane is partitioned into congruent equilateral triangles such that any two triangles have no point in common, or share one vertex, or share one edge. Does there exist a disk containing exactly 1988 vertices of these triangles?

A bounded sequence $\{x_n\}$ satisfies $x_n+x_{n+1}\ge2x_{n+2}$ for every $n\ge1$. Must it converge?

Suppose an acute triangle $ABC$ exists such that $\tan A,\tan B,\tan C$ are the three roots of $x^3+px^2+qx+p=0$, where $q\ne1$. Prove that $p\le-3\sqrt3$ and $q\gt 1$.

In space, let $a,b,c$ be three pairwise skew lines. Prove that $a,b,c$ have a common perpendicular if and only if the composition of the three symmetries about $a,b,c$, in that order, is also a symmetry about a line.