AT-OEMO-FED1 2008

What is the remainder of the number $1 \binom{2008}{0 }+2\binom{2008}{1}+ ...+2009\binom{2008}{2008}$ when divided by $2008$ ?

Given $a \in R^{+}$ and an integer $n > 4$ determine all n-tuples ($x_1, ...,x_n$) of positive real numbers that satisfy the following system of equations: $\begin {cases} x_1x_2(3a-2x_3) = a^3\\ x_2x_3(3a-2x_4) = a^3\\ ...\\ x_{n-2}x_{n-1}(3a-2x_n) = a^3\\ x_{n-1}x_n(3a-2x_1) = a^3 \\ x_nx_1(3a-2x_2) = a^3 \end {cases}$ .

Let $p > 1$ be a natural number. Consider the set $F_p$ of all non-constant sequences of non-negative integers that satisfy the recursive relation $a_{n+1} = (p+1)a_n - pa_{n-1}$ for all $n > 0$ . Show that there exists a sequence ( $a_n$ ) in $F_p$ with the property that for every other sequence ( $b_n$ ) in $F_p$ , the inequality $a_n \le b_n$ holds for all $n$ .

In a triangle $ABC$ let $E$ be the midpoint of the side $AC$ and $F$ the midpoint of the side $BC$ . Let $G$ be the foot of the perpendicular from $C$ to $AB$ . Show that $\vartriangle EFG$ is isosceles if and only if $\vartriangle ABC$ is isosceles.