A parallelogram is given in the Cartesian coordinate system. If the corner points of the parallelogram are connected to the midpoints of adjacent sides using lines, then the eight connecting lines form an octagon.
It must be proven that its area is one sixth of the parallelogram's area.

A gardener wants to spread 25 roses over an area so that there are 5 roses in each of 15 straight rows. The roses should be arranged rotationally symmetrically so that more than 3/4 of them are less than half as far from the center of symmetry as the outermost ones and that the center of symmetry itself remains unplanted. How is such an arrangement possible?

Prove: If a is an irrational number, then the function y = cos (ax) + cos x is not periodic.
Is it possible to graphically represent or calculate this fact using an example?

If a point is chosen in the Euclidean plane, then it is red or black.
It must be proven that there is then an equilateral triangle with corners of the same color.

An oddly shaped Christmas package is a collection of centrally stacked cube-shaped boxes with ever smaller sides. From bottom to top, the edge lengths of the stacked cubes run like the sequence of the reciprocals of the sequence of the square roots of the natural numbers. How high is the package? How much paper is needed to wrap the package? What is the total volume enclosed by all the packages?