NL-NWO 1967

In this exercise we only consider convex quadrilaterals.

(a) For such a quadrilateral $ABCD$ , determine the set of points $P$ contained within that quadrilateral for which $PA$ and $PC$ divide the quadrilateral into two pieces of equal areas.

(b) Prove that there is a point $P$ inside such a quadrilateral, such that the triangles $PAB$ and $PCD$ have equal areas, as well as the triangles $PBC$ and $PAD$ .

(c) Find out which quadrilaterals $ABCD$ contains a point $P$ , so that the triangles $PAB$ , $PBC$ , $PCD$ and $PDA$ have equal areas.

Consider arithmetic sequences where all terms are natural numbers. If the first term of such a sequence is $1$ , prove that that sequence contains infinitely many terms that are the cube of a natural number. Give an example of such a sequence in which no term is the cube of a natural number and show the correctness of this example.

The convex pentagon $ABC DE$ is given, such that $AB,BC,CD$ and $DE$ are parallel to one of the diagonals. Prove that this also applies to $EA$ .

The following applies: $$a, b, c, d > 0 , a + b < c + d$$ Prove that $$ac + bd > ab.$$

Consider rows of the form: $[x], [2x], [3x], ...$ Proof that, if $N \in N$ does not occur in the sequence $([n x])$ , then there is an $n \in N$ with $n - 1 < \frac{N}{x}< n -\frac{1}{x}$ Prove that, for $x, y \notin Q$ : $\frac{1}{x}+\frac{1}{y} = 1$ , then each $N \in N$ term is either of $([nx])$ or of $([ny])$ .