PL-OM 1996

Find all pairs $(n,r)$ with $n$ a positive integer and $r$ a real such that $2x^2+2x+1$ divides $(x+1)^n - r$ .

$ABCD$ is a tetrahedron with $\angle BAC = \angle ACD$ and $\angle ABD = \angle BDC$ . Show that $AB = CD$ .

Let $p(k)$ be the smallest prime not dividing $k$ . Put $q(k) = 1$ if $p(k) = 2$ , or the product of all primes $< p(k)$ if $p(k) > 2$ . Define the sequence $x_0, x_1, x_2, ...$ by $x_0 = 1$ , $x_{n+1} = \frac{x_np(x_n)}{q(x_n)}$ . Find all $n$ such that $x_n = 111111$