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What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

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Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

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Incircles

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

Sweeping Satellite

Stage: 5 Challenge Level: Challenge Level:1

Hint 1: Gravitational potential energy is the integral of the gravitational force, and evaluates to a negative number, theoretically being zero when infinitely far away.

Hint 2: In the Newtonian gravitational field equation, it can be shown that $GM = gR^2$. That assumes a spherical body of course.