AT-OEMO-FED1 2004

Find all quadruples $(a, b, c, d)$ of real numbers such that $$a + bcd = b + cda = c + dab = d + abc.$$

A convex hexagon $ABCDEF$ with $AB = BC = a, CD = DE = b, EF = FA = c$ is inscribed in a circle. Show that this hexagon has three (pairwise disjoint) pairs of mutually perpendicular diagonals.

For natural numbers $a, b$ , define $Z(a,b)=\frac{(3a)!\cdot (4b)!}{a!^4 \cdot b!^3}$ .(a) Prove that $Z(a, b)$ is an integer for $a \leq b$ .(b) Prove that for each natural number $b$ there are infinitely many natural numbers a such that $Z(a, b)$ is not an integer.[/list]

Each of the $2N = 2004$ real numbers $x_1, x_2, \ldots , x_{2004}$ equals either $\sqrt 2 -1$ or $\sqrt 2 +1$ . Can the sum $\sum_{k=1}^N x_{2k-1}x_2k$ take the value $2004$ ? Which integral values can this sum take?