DK-GEORGMOHR 2020

The figure shows $9$ circles connected by $12$ lines. Georg must colour each circle either red or blue. He gets one point for each line connecting circles with different colours. How many points can he at most achieve?

A quadrilateral is cut from a piece of gift wrapping paper, which has equally wide white and gray stripes. The grey stripes in the quadrilateral have a combined area of $10$ . Determine the area of the quadrilateral.

Which positive integers satisfy the following three conditions? a) The number consists of at least two digits. b) The last digit is not zero. c) Inserting a zero between the last two digits yields a number divisible by the original number.

Identical rectangular cardboard pieces are handed out to $30$ students, one to each. Each student cuts (parallel to the edges) his or her piece into equally large squares. Two different students' squares do not necessarily have the same size. After all the cutting it turns out that the total number of squares is a prime. Prove that the original cardboard pieces must have been quadratic.

Alma places spies on some of the squares on a $2020\times 2020$ game board. Now Bertha secretly chooses a quadradic area consisting of $1020 \times 1020$ squares and tells Alma which spies are standing on a square in the secret quadradic area. At least how many spies must Alma have placed in order for her to determine with certainty which area Bertha has chosen?