Explaining, convincing and proving

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

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

Show that the edges $AD$ and $BC$ of a tetrahedron $ABCD$ are mutually perpendicular if and only if $AB^2 +CD^2 = AC^2+BD^2$. This problem uses the scalar product of two vectors.

Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.

Sort these mathematical propositions into a series of 8 correct statements.