AT-OEMO-FED1 2007

In a quadratic table with $2007$ rows and $2007$ columns is an odd number written in each field. For $1\leq i\leq2007$ is $Z_i$ the sum of the numbers in the $i$ -th row and for $1\leq j\leq2007$ is $S_j$ the sum of the numbers in the $j$ -th column. $A$ is the product of all $Z_i$ and $B$ the product of all $S_j$ . Show that $A+B\neq0$

For every positive integer $n$ determine the highest value $C(n)$ , such that for every $n$ -tuple $(a_1,a_2,\ldots,a_n)$ of pairwise distinct integers

$(n + 1)\sum_{j = 1}^n a_j^2 - \left(\sum_{j = 1}^n a_j\right)^2\geq C(n)$

Let $M(n )=\{-1,-2,\ldots,-n\}$ . For every non-empty subset of $M(n )$ we consider the product of its elements. How big is the sum over all these products?

Let $n > 4$ be a non-negative integer. Given is the in a circle inscribed convex $n$ -gon $A_0A_1A_2\dots A_{n - 1}A_n$ $(A_n = A_0)$ where the side $A_{i - 1}A_i = i$ (for $1 \le i \le n$ ). Moreover, let $\phi_i$ be the angle between the line $A_iA_{i + 1}$ and the tangent to the circle in the point $A_i$ (where the angle $\phi_i$ is less than or equal $90^o$ , i.e. $\phi_i$ is always the smaller angle of the two angles between the two lines). Determine the sum $\Phi = \sum_{i = 0}^{n - 1} \phi_i$ of these $n$ angles.