Explaining, convincing and proving

problem
Long short

Age

14 to 16

Challenge level

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
problem
Mod 3

Age

14 to 16

Challenge level

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
problem
Our ages

Age

14 to 16

Challenge level

I am exactly n times my daughter's age. In m years I shall be ... How old am I?
problem
DOTS division

Age

14 to 16

Challenge level

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
problem
There's a limit

Age

14 to 18

Challenge level

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?
problem
Diophantine n-tuples

Age

14 to 16

Challenge level

Can you explain why a sequence of operations always gives you perfect squares?
problem
Pythagorean golden means

Age

16 to 18

Challenge level

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.
problem
Euler's squares

Age

14 to 16

Challenge level

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...
problem
Three cubes

Age

14 to 16

Challenge level

Can you work out the dimensions of the three cubes?
problem
Orthogonal circle

Age

16 to 18

Challenge level

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.