In the decimal system, we are looking for all natural numbers with at most six digits that only swap the order of the digits when multiplied by 2, 3, ..., 6.

In a plane, equilateral triangles D(i) with side lengths a(i)= 2*i−1, i = 1; 2; 3; ... are arranged along a straight line g in such a way that the "right" corner point of triangle D(k) coincides with the "left" corner point of triangle D(k+1) and that the third corner points all lie in the same half-plane generated by g. Determine the curve/function on which the third corner points lie!

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?