NL-NWO 1985

For some $p$ , the equation $x^3 + px^2 + 3x - 10 = 0$ has three real solutions $a,b,c$ such that $c - b = b - a > 0$ . Determine $a,b,c,$ and $p$ .

Among the numbers $11n + 10^{10}$ , where $1 \le n \le 10^{10}$ is an integer, how many are squares?

In a factory, square tables of $40 \times 40$ are tiled with four tiles of size $20 \times 20$ . All tiles are the same and decorated in the same way with an asymmetric pattern such as the letter $J$ . How many different types of tables can be produced in this way?

A convex hexagon $ABCDEF$ is such that each of the diagonals $AD,BE,CF$ divides the hexagon into two parts of equal area. Prove that these three diagonals are concurrent.