NL-NWO 1974

A convex quadrilateral with area $1$ is divided into four quadrilaterals divided by connecting the midpoints of the opposite sides. Prove that each of those four quadrilaterals has area $< \frac38$ .

$n>2$ numbers, $x_1, x_2, ..., x_n$ are odd . Prove that $4$ divides $$x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 -n.$$

Proove that in every five positive numbers there is a pair, say $a,b$ , for which $$\left| \frac{1}{a+25}- \frac{1}{b+25}\right| <\frac{1}{100}.$$

For which $n$ is $n^4+6n^3+11n^2+3n+31$ a perfect square?

For every $n \in N$ , is it possible to make a figure consisting of $n+1$ points, where $n$ points lie on one line and one point is not on that line, so that each pair of those points is an integer distance from each other?