NL-NWO 2001

In a tournament, every team plays exactly once against every other team. One won match earns $3$ points for the winner and $0$ for the loser. With a draw both teams receive $1$ point each. At the end of the tournament it appears that all teams together have achieved $15$ points. The last team on the final list scored exactly $1$ point. The second to last team has not lost a match. a) How many teams participated in the tournament? b) How many points did the team score in second place in the final ranking?

The function f has the following properties : $f(x + y) = f(x) + f(y) + xy$ for all real $x$ and $y$ $f(4) = 10$ Calculate $f(2001)$ .

A wooden beam $EFGH$ $ABCD$ is with three cuts in $8$ smaller ones sawn beams. Each cut is parallel to one of the three pair of opposit sides. Each pair of saw cuts is shown perpendicular to each other. The smaller bars at the corners $A, C, F$ and $H$ have a capacity of $9, 12, 8, 24$ respectively.(The proportions in the picture are not correct!!). Calculate content of the entire bar.

$A$ $B$ $C$ $E$ $F$ $G$ $H$

The function is given $f(x) = \frac{2x^3 -6x^2 + 13x + 10}{2x^2 - 9x}$ .

Determine all positive integers $x$ for which $f(x)$ is an integer

If you take a subset of $4002$ numbers from the whole numbers $1$ to $6003$ , then there is always a subset of $2001$ numbers within that subset with the following property: If you order the $2001$ numbers from small to large, the numbers are alternately even and odd (or odd and even). Prove this.