Explaining, convincing and proving

problem

The Pillar of Chios

Age

14 to 16

Challenge level

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
problem
Favourite

Napkin

Age

14 to 16

Challenge level

A napkin is folded so that a corner coincides with the midpoint of an opposite edge. Investigate the three triangles formed.
problem

Cyclic Quad Jigsaw

Age

14 to 16

Challenge level

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?
problem
Favourite

Angle Trisection

Age

14 to 16

Challenge level

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
problem

AMGM

Age

14 to 16

Challenge level

Can you use the diagram to prove the AM-GM inequality?
problem
Favourite

Lens Angle

Age

14 to 16

Challenge level

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.
problem
Favourite

Hexy-Metry

Age

14 to 16

Challenge level

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
problem
Favourite

Squirty

Age

14 to 16

Challenge level

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.
problem
Favourite

Far Horizon

Age

14 to 16

Challenge level

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
problem
Favourite

Harmonic Triangle

Age

14 to 16

Challenge level

Can you see how to build a harmonic triangle? Can you work out the next two rows?