Reasoning, convincing and proving

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?

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.

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 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.

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.