Explaining, convincing and proving

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

What can you say about the common difference of an AP where every term is prime?

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

Can you prove that in every tetrahedron there is a vertex where the three edges meeting at that vertex have lengths which could be the sides of a triangle?

This is a beautiful result involving a parabola and parallels.

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

There are many different methods to solve this geometrical problem - how many can you find?