Creating and manipulating expressions and formulae

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

Can you use the diagram to prove the AM-GM inequality?

Can you explain what's happening with these numbers?

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

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