NL-NWO 2012

Let $a, b, c$ , and $d$ be four distinct integers. Prove that $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ is divisible by $12$ .

We number the columns of an $n\times n$ -board from $1$ to $n$ . In each cell, we place a number. This is done in such a way that each row precisely contains the numbers $1$ to $n$ (in some order), and also each column contains the numbers $1$ to $n$ (in some order). Next, each cell that contains a number greater than the cell's column number, is coloured grey. In the figure below you can see an example for the case $n = 3$ .

$1$ $2$ $3$ $3$ $1$ $2$ $1$ $2$ $3$ $2$ $3$ $1$

(a) Suppose that $n = 5$ . Can the numbers be placed in such a way that each row contains the same number of grey cells? (b) Suppose that $n = 10$ . Can the numbers be placed in such a way that each row contains the same number of grey cells?

Determine all pairs $(p,m)$ consisting of a prime number $p$ and a positive integer $m$ , for which $p^3 + m(p + 2) = m^2 + p + 1$ holds.

We are given an acute triangle $ABC$ and points $D$ on $BC$ and $E$ on $AC$ such that $AD$ is perpendicular to $BC$ and $BE$ is perpendicular to $AC$ . The intersection of $AD$ and $BE$ is called $H$ . A line through $H$ intersects line segment $BC$ in $P$ , and intersects line segment $AC$ in $Q$ . Furthermore, $K$ is a point on $BE$ such that $PK$ is perpendicular to $BE$ , and $L$ is a point on $AD$ such that $QL$ is perpendicular to $AD$ . Prove that $DK$ and $EL$ are parallel.

$A$ $B$ $C$ $D$ $E$ $H$ $K$ $L$ $P$ $Q$

The numbers $1$ to $12$ are arranged in a sequence. The number of ways this can be done equals $12 \times11 \times 10\times ...\times 1$ . We impose the condition that in the sequence there should be exactly one number that is smaller than the number directly preceding it. How many of the $12 \times11 \times 10\times ...\times 1$ sequences satisfy this condition?