The following problem is old and quite challenging. I found it in a publication from 1959. I have the source and my own solution. I was able to solve it quite laboriously a few years ago using MC14 – but not recently with Maple. Therefore, I'm interested in instructive Maple solutions, just as previous puzzle solutions here have helped me further in using this software. I thank you for that.

A wolf observes a goat. It is grazing in a meadow, tied to a rope. Naturally, he wants to catch the goat and considers whether his starting position is favorable. He estimates that he and the goat will be moving at the same speed at the beginning of the hunt and throughout the pursuit, and that the goat will constantly hold the rope taut to maintain its distance for escape reasons. Now, as he approaches the goat at a constant top speed, the same speed as the goat is fleeing in a circular arc, he asks himself: Where do I need to start from to guarantee success? Are there starting positions that rule out a successful hunt? What happens if I manage to sneak into the grazed area first?

In the plane, the concentric circles k1 and k2 are given with center M. Circle k1 is the unit circle (radius = 1), and k2 has twice the area of ​​k1. From the outside, five congruent circles k3 are placed tangent to k1, each with a radius r yet to be calculated. Prove that the circles k3 can be arranged such that any two adjacent circles k3 and k2 can have a common intersection point, and these intersection points form the vertices of a regular pentagon. The radius r is to be calculated exactly (no approximation) as a term.

Prove:
If a is an irrational number, then the function
y(x) = cos(a*x) + cosx is not periodic.

Further question:
Since every irrational number can be represented as the limit of a sequence of rational numbers and all software "only" works with finite representations of numbers, it would be interesting to know at what number length for a this becomes in examples apparent in the plot for y(x).

In the attached file, the trigonometric term (2, term) is transformed into a term (3, term1) consisting of radicals. Is there a Maple procedure that can be used to reverse this process? Given an algebraic term (e.g., consisting of radicals, powers, etc.), under what conditions can it be transformed into a trigonometric form (not a Fourier series) in the sense of (3) according to (2)?test.mw

Download test.mw

...flies around in the spatial Cartesian coordinate system and impacts the coordinate planes xy, xz, and yz with exactly one impact. This creates circular impressions on the coordinate planes in this order, with the radii r1, r2, and r3, which are assumed to be known. From these, the sphere's radius r and the coordinates of its center can be determined?