NL-NWO 2013

In a table consisting of $n$ by $n$ small squares some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of $n$ ?

Find all triples $(x, y, z)$ of real numbers satisfying: $x + y - z = -1$ , $x^2 - y^2 + z^2 = 1$ and $- x^3 + y^3 + z^3 = -1$

The sides $BC$ and $AD$ of a quadrilateral $ABCD$ are parallel and the diagonals intersect in $O$ . For this quadrilateral $|CD| =|AO|$ and $|BC| = |OD|$ hold. Furthermore $CA$ is the angular bisector of angle $BCD$ . Determine the size of angle $ABC$ .

$A$ $B$ $C$ $D$ $O$

Attention: the figure is not drawn to scale.

For a positive integer n the number $P(n)$ is the product of the positive divisors of $n$ . For example, $P(20) = 8000$ , as the positive divisors of $20$ are $1, 2, 4, 5, 10$ and $20$ , whose product is $1 \cdot 2 \cdot 4 \cdot 5 \cdot 10 \cdot 20 = 8000$ . (a) Find all positive integers $n$ satisfying $P(n) = 15n$ . (b) Show that there exists no positive integer $n$ such that $P(n) = 15n^2$ .

The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$ If one writes down the number $S$ , how often does the digit ` $5$ ' occur in the result?