NL-NWO 1977

Solve, for integers $x$ and $y$: $$2x^2y = (x+2)^2(y + 1),$$ provided that $(x+2)^2(y + 1)> 1000$.

Four masts stand on a flat horizontal piece of land at the vertices of a square $ABCD$. The height of the mast on $A$ is $7$ meters, of the mast on $B$ $13$ meters, and of the mast on $C$ $15$ meters. Within the square there is a point $P$ on the ground equidistant from each of the tops of these three masts. (a) What length must the sides of the square be at least for this to be possible? (b) The distance from $P$ to the top of the mast on $D$ is equal to the distance fro m $P$ to each of the tops of the three other masts. Calculate the height of the mast at $D$.

From each set $\{a_1,a_2,...,a_7\} \subset Z$ one can choose a number of elements whose sum is a multiple of $7$.

There are an even number of points in a plane. No three of them lie on one straight line. Half of the points are red, the other half are blue. Prove that there exists a connecting line of a red and a blue point such that in each of the half-planes bounded by that line the number of red points is equal to the number of blue points.