AT-OEMO-REG 2001

Let $n$ be an integer. We consider $s (n)$ , the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$ . So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?

Find all real solutions to the equation $$(x+1)^{2001}+(x+1)^{2000}(x-2)+(x+1)^{1999}(x-2)^2+...+(x+1)^2(x-2)^{1999}+(x+1)^{2000}(x-2)+(x+1)^{2001}=0$$

In a convex pentagon $ABCDE$ , the area of the triangles $ABC, ABD, ACD$ and $ADE$ are equal and have the value $F$ . What is the area of the triangle $BCE$ ?

Let $A_o =\{1, 2\}$ and for $n> 0, A_n$ results from $A_{n-1}$ by adding the natural numbers to $A_{n-1}$ which can be represented as the sum of two different numbers from $A_{n-1}$ . Let $a_n = |A_n |$ be the number of numbers in $A_n$ . Determine $a_n$ as a function of $n$ .