PL-OM 2008

A function $f: R^3\rightarrow R$ for all reals $a,b,c,d,e$ satisfies a condition:

$$f(a,b,c)+f(b,c,d)+f(c,d,e)+f(d,e,a)+f(e,a,b)=a+b+c+d+e$$

Show that for all reals $x_1,x_2,\ldots,x_n$ ( $n\geq 5$ ) equality holds:

$$f(x_1,x_2,x_3)+f(x_2,x_3,x_4)+\ldots +f(x_{n-1},x_n,x_1)+f(x_n,x_1,x_2)=x_1+x_2+\ldots+x_n$$

In a convex pentagon $ABCDE$ in which $BC=DE$ following equalities hold:

$$\angle ABE =\angle CAB =\angle AED-90^{\circ}, \angle ACB=\angle ADE$$

Show that $BCDE$ is a parallelogram.

Let $R$ be a parallelopiped. Let us assume that areas of all intersections of $R$ with planes containing centers of three edges of $R$ pairwisely not parallel and having no common points, are equal. Show that $R$ is a cuboid.