ZA-SAMO 2000

A number $x_n$ of the form 10101...1 has $n$ ones. Find all $n$ such that $x_n$ is prime.

Solve for $x$ , given $36x^4 + 36x^3 - 7x^2 - 6x + 1 = 0$ .

Let $c \geq 1$ be an integer, and define the sequence $a_1,\ a_2,\ a_3,\ \dots$ by $$\begin{aligned} a_1 & = 2, a_{n + 1} & = ca_n + \sqrt{\left(c^2 - 1\right)\left(a_n^2 - 4\right)}\textrm{ for }n = 1,2,3,\dots\ . \end{aligned}$$ Prove that $a_n$ is an integer for all $n$ .

$ABCD$ is a square of side 1. $P$ and $Q$ are points on $AB$ and $BC$ such that $\widehat{PDQ} = 45^{\circ}$ . Find the perimeter of $\Delta PBQ$ .

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ (where $\mathbb{Z}$ is the set of all integers) such that $$2000f(f(x)) - 3999f(x) + 1999x = 0\textrm{ for all }x \in \mathbb{Z}.$$

Let $A_n$ be the number of ways to tile a $4 \times n$ rectangle using $2 \times 1$ tiles. Prove that $A_n$ is divisible by 2 if and only if $A_n$ is divisible by 3.