NL-NWO 1966

A chord $AB$ is drawn in a circle, with center $M$ and radius $r$ , that the two diameters which divide the largest arc $AB$ into three equal parts also divide the chord $AB$ into three equal parts. Express the length of that chord in terms of $r$ .

For all $n$ , $t_{n+1} = 2(t_n)^2 - 1$ . Prove that gcd $(t_n,t_m) = 1$ if $n \ne m$ .

How many natural numbers are there whose square is a thirty-digit number which has the following curious property: If that thirty-digit number is divided from left to right into three groups of ten digits, then the numbers given by the middle group and the right group formed numbers are both four times the number formed by the left group?

A rectangular piece of paper is divided into square cells by lines parallel to the sides of the rectangle. $n$ (horizontal) rows of $m$ cells have emerged and $m$ (vertical) columns of $n$ cells have also been formed. There is a number in each cell. Find the largest number in each of the $n$ rows. The smallest maxima of those $n$ rows is called $A$ . We also look for the smallest number in each of the $m$ columns. The largest minima of those $m$ columns is called $B$ . Prove that $A$ is greater than or equal to $B$ . Can you give a simple example where $A = B$ ?

The image that maps $x$ to $1 - x$ is called complement, the image that maps $x$ to $\frac{1}{x}$ is called invert. Two numbers $x$ and $y$ are called related if they can be transferred into each other by means of complementationand/or inversion. A family is a collection of numbers where every two elements are related. Determine the maximum size $n$ of such a family. Show that the number line can be divided into $n$ parts, such that each of those $n$ parts contains exactly one number from each $n$ -number family.