DE-MO 2001

Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression

Determine the maximum possible number of points you can place in a rectangle with lengths $14$ and $28$ such that any two of those points are more than $10$ apart from each other.

Wiebke and Stefan play the following game on a rectangular sheet of paper. They start with a rectangle with $60$ rows and $40$ columns and cut it in turns into smaller rectangles. The cuttings must be made along the gridlines, and a player in turn may cut only one smaller rectangle. By that, Stefan makes only vertical cuts, while Wiebke makes only horizontal cuts. A player who cannot make a regular move loses the game. (a) Who has a winning strategy if Stefan makes the first move? (b) Who has a winning strategy if Wiebke makes the first move?

In how many ways can the "Nikolaus' House" (see the picture) be drawn? Edges may not be erased nor duplicated, and no additional edges may be drawn.

The Fibonacci sequence is given by $x_1 = x_2 = 1$ and $x_{k+2} = x_{k+1} + x_k$ for each $k \in N$ . (a) Prove that there are Fibonacci numbes that end in a $9$ in the decimal system. (b) Determine for which $n$ can a Fibonacci number end in $n$ $9$ -s in the decimal system.