KURSCHAK 2006

Is there a set $S\subset\mathbb{R}^3$ of $2006$ points such that not all its points are coplanar, no three of the points are collinear, and for any $A,B\in S$ we can find points $C,D\in S$ for which $AB||CD$ ?

Let $a,t,n$ be positive integers such that $a\le n$ . Consider the subsets of $\{1,2,\dots,n\}$ whose any two elements differ by at least $t$ . Prove that the number of such subsets not containing $a$ is at most $t^2$ times the number of those that do contain $a$ .

We deal $n-1$ cards in some way to $n$ people sitting around a table. From then on, in one move a person with at least $2$ cards gives one card to each of his/her neighbors. Prove that eventually a state will be reached where everyone has at most one card.