NL-NWO 1981

$f(x) = [x] + [2x] + [3x] + [4x] + [5x] + [6x]$ . What values does $f$ take?

Given is the equilateral triangle $ABC$ with center $M$ . On $CA$ and $CB$ the respective points $D$ and $E$ lie such that $CD = CE$ . $F$ is such that $DMFB$ is a parallelogram. Prove that $\vartriangle MEF$ is equilateral.

We want to split the set of natural numbers from $1$ to $3n$ , where $n$ is a natural number, into $n$ mutually disjoint sets $\{x,y,z\}$ of three elements such that always holds: $x + y = 3z$ . Is this possible for : a) $n = 5$ ? b) $n=10$ ? In both cases, provide either such a split or proof that such a split is not possible.

A wire figure is held in different ways in a bundle of parallel light rays, so that different shadow figures are created in a plane perpendicular to the light rays. In this way one can form: (a) an isosceles triangle; (b) an isosceles triangle with altitude from the apex; (c) a rectangle containing an isosceles triangle; (d) a rhombus with one diagonal. The wire figure consists of eight straight pieces of iron wire, with each piece connected to both ends are attached to at least one other piece. Determine a figure corresponding to the above description is satisfactory, and indicate the direction of the light rays at which the shadow figures (a) to (d) arise.