Explaining, convincing and proving

This problem challenges you to find cubic equations which satisfy different conditions.

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

Find all the solutions to the this equation.

What can you say about the common difference of an AP where every term is prime?

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

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.

This is a beautiful result involving a parabola and parallels.

Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?