Reasoning, convincing and proving

problem
Summats clear

Age

16 to 18

Challenge level

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.
problem
Napoleon's hat

Age

16 to 18

Challenge level

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?
problem
Shape and territory

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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

Age

14 to 16

Challenge level

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?
problem
OK! Now prove it

Age

16 to 18

Challenge level

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

Age

14 to 16

Challenge level

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

Age

14 to 16

Challenge level

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

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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?