NL-NWO 1972

Prove that for every $n \in N$ , $n > 6$ , every equilateral triangle can be divided into $n$ pieces, which are also equilateral triangles.

Prove that there exists exactly one function $ƒ$ which is defined for all $x \in R$ , and for which holds: $\bullet$ $x \le y \Rightarrow f(x) \le f(y)$ , for all $x, y \in R$ , and $\bullet$ $f(f(x)) = x$ , for all $x \in R$ .

$ABCD$ is a regular tetrahedron. The points $P,Q,R$ and $S$ lie outside this tetrahedron in such a way that $ABCP$ , $ABDQ$ , $ACDR$ and $BCDS$ are regular tetrahedra. Prove that the volume of the tetrahedron $PQRS$ is less than the sum of the volumes of $ABCP$ , $ABDQ$ , $ACDR$ , $BCDS$ and $ABCD$ .

On a circle with radius $1$ the points $A_1, A_2,..., A_n$ lie such that every arc $A_iA_{i+i}$ has length $\frac{2\pi}{n}= a$ . Given that there exists a set $V$ consisting of $k$ of these points ( $k < n$ ), which has the property that each of the arc lengths $a$ , $2a$ $,...$ , $(n- 1)a$ can be obtained in exactly one way be taken as the length of an arc traversed in a positive sense, beginning and ending in a point of $V$ . Express $n$ in terms of $k$ and give the set $V$ for the case $n = 7$ .

Given is an acute-angled triangle $ABC$ with angles $\alpha$ , $\beta$ and $\gamma$ . On side $AB$ lies a point $P$ such that the line connecting the feet of the perpendiculars from $P$ on $AC$ and $BC$ is parallel to $AB$ . Express the ratio $\frac{AP}{BP}$ in terms of $\alpha$ and $\beta$ .