A bundle of planes is here defined as the set of all planes in Cartesian (x, y, z)-space that intersect along the same straight line. This line is commonly referred to as the axis of the bundle. Is there an efficient method in Maple to model a bundle of planes in a simple, symbolic way?
The background to my request for advice is the following problem (for those interested in the puzzle), dating back to 1965:
Let a surface be given in Euclidean space. Suppose that every planar section of the surface is a circle, provided the section contains more than one point. Prove that the surface is a sphere.

edited:

Simple deductions are sufficient to solve the puzzle. Complex calculations are not necessary.

A Blog Post reminded me of an old problem from my Mathcad-days:
Given a torus and two points, P and Q, on its surface, calculate and graphically represent the shortest path from P to Q.

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Can Mathcad worksheets (MC14) be converted into functional worksheets for Maple?

In the attached file, I want to represent a mixed polynomial in x, y, and z as the sum of squares from bracketed expressions. I know the result. But how do I do this in Maple? The help text didn't help me :-( .

test.mw

The following statement is to be examined:

For every natural number n, there exists a natural number m such that (sqrt(2)-1)^n = sqrt(m+1)-sqrt(m).

I would like to calculate a definite integral in the attached file. Doing so numerically is straightforward. However, a symbolic approach initially fails due to three points of discontinuity in the integrand. Yet, these points can be remedied by taking limits. How can the domain in Maple be extended using these limits to achieve continuity across the entire domain?

testint.mw