Creating and manipulating expressions and formulae

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

Relate these algebraic expressions to geometrical diagrams.

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

Can you hit the target functions using a set of input functions and a little calculus and algebra?

Derive an equation which describes satellite dynamics.

Can you fit polynomials through these points?

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

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.

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?