Explaining, convincing and proving

Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

Here the diagram says it all. Can you find the diagram?

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.

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

An inequality involving integrals of squares of functions.

Follow the hints and prove Pick's Theorem.

Show that it is rare for a ratio of ratios to be rational.