NL-NWO 2003

A Pythagorean triangle is a right triangle whose three sides are integers. The best known example is the triangle with rectangular sides $3$ and $4$ and hypotenuse $5$ . Determine all Pythagorean triangles whose area is twice the perimeter.

Two squares with side $12$ lie exactly on top of each other. One square is rotated around a corner point through an angle of $30$ degrees relative to the other square. Determine the area of the common piece of the two squares.

$30^\circ$

Determine all positive integers $n$ that can be written as the product of two consecutive integers and as well as the product of four consecutive integers numbers. In the formula: $n = a (a + 1) = b (b + 1) (b + 2) (b + 3)$ .

In a circle with center $M$ , two chords $AC$ and $BD$ intersect perpendicularly. The circle of diameter $AM$ intersects the circle of diameter $BM$ besides $M$ also in point $P$ . The circle of diameter $BM$ intersects the circle with diameter $CM$ besides $M$ also in point $Q$ . The circle of diameter $CM$ intersects the circle of diameter $DM$ besides $M$ also in point $R$ . The circle of diameter $DM$ intersects the circle of diameter $AM$ besides $M$ also in point $S$ . Prove that quadrilateral $PQRS$ is a rectangle.

$A$ $B$ $C$ $D$ $M$ $P$ $Q$ $R$ $S$

There are a number of cards on a table. A number is written on each card. The "pick and replace" operation involves the following: two random cards are taken from the table and replaced by one new card. If the numbers $a$ and $b$ appear on the two packed cards, the number $a + b + ab$ is set on the new card. If we start with ten cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ and $10$ respectively, what value(s) can the number have that "grab and replace" nine times is on the only card still on the table? Prove your answer