Explaining, convincing and proving

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Four jewellers share their stock. Can you work out the relative values of their gems?

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

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. Does this always work? Can you prove or disprove this conjecture?

The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

Do points P and Q lie inside, on, or outside this circle?