NL-NWO 2021

Niek has $16$ square cards that are yellow on one side and red on the other. He puts down the cards to form a $4 \times 4$ -square. Some of the cards show their yellow side and some show their red side. For a colour pattern he calculates the monochromaticity as follows. For every pair of adjacent cards that share a side he counts $+1$ or $-1$ according to the following rule: $+1$ if the adjacent cards show the same colour, and $-1$ if the adjacent cards show different colours. Adding this all together gives the monochromaticity (which might be negative). For example, if he lays down the cards as below, there are $15$ pairs of adjacent cards showing the same colour, and $9$ such pairs showing different colours.

The monochromaticity of this pattern is thus $15 \cdot (+1) + 9 \cdot (-1) = 6$ . Niek investigates all possible colour patterns and makes a list of all possible numbers that appear at least once as a value of the monochromaticity. That is, Niek makes a list with all numbers such that there exists a colour pattern that has this number as its monochromaticity.

(a) What are the three largest numbers on his list? (Explain your answer. If your answer is, for example, $12$ , $9$ and $6$ , then you have to show that these numbers do in fact appear on the list by giving a colouring for each of these numbers, and furthermore prove that the numbers $7$ , $8$ , $10$ , $11$ and all numbers bigger than $12$ do not appear.)

(b) What are the three smallest (most negative) numbers on his list?

(c) What is the smallest positive number (so, greater than $0$ ) on his list?

We consider sports tournaments with $n \ge 4$ participating teams and where every pair of teams plays against one another at most one time. We call such a tournament balanced if any four participating teams play exactly three matches between themselves. So, not all teams play against one another. Determine the largest value of $n$ for which a balanced tournament with $n$ teams exists.

A frog jumps around on the grid points in the plane, from one grid point to another. The frog starts at the point $(0, 0)$ . Then it makes, successively, a jump of one step horizontally, a jump of $2$ steps vertically, a jump of $3$ steps horizontally, a jump of $4$ steps vertically, et cetera. Determine all $n > 0$ such that the frog can be back in $(0, 0)$ after $n$ jumps.

In triangle $ABC$ we have $\angle ACB = 90^o$ . The point $M$ is the midpoint of $AB$ . The line through $M$ parallel to $BC$ intersects $AC$ in $D$ . The midpoint of line segment $CD$ is $E$ . The lines $BD$ and $CM$ are perpendicular. (a) Prove that triangles $CME$ and $ABD$ are similar. (b) Prove that $EM$ and $AB$ are perpendicular.

$A$ $B$ $C$ $D$ $E$ $M$

Be aware: the figure is not drawn to scale.

We consider an integer $n > 1$ with the following property: for every positive divisor $d$ of $n$ we have that $d + 1$ is a divisor of $n + 1$ . Prove that $n$ is a prime number.