Creating and manipulating expressions and formulae

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

Can you prove that twice the sum of two squares always gives the sum of two squares?

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.