Conjecturing and generalising

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?

Generalise this inequality involving integrals.

Can you make matrices which will fix one lucky vector and crush another to zero?

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.

Can you find some Pythagorean Triples where the two smaller numbers differ by 1?

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.