Reasoning, convincing and proving

Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Do you have enough information to work out the area of the shaded quadrilateral?

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?