Following CMS Open Evening on 15th October, this is a hint to the problem and to provide those that attended the open evening a chance and some time to submit solutions before it is released.
Due to this, the solution for both problems will be delayed.
Apologies for the late notice and thank you to Ishaan Agarwal for submitting a creditable solution before the deadline.
We would like to emphasise that partial solutions are acceptable and we cannot wait to see your responses!
a) It is worthwhile to consider radii and lengths relating to the equilateral triangle. The angles in an equilateral triangle allow for exact trigonometric values.
bi*) The relationship between the radius of the sphere (
) and the side length of a tetrahedron (
) is 
The relationship between the height of the tetrahedron (
) and the side length is 
Can you continue from here using similar logic from 
?
bii*) We will admit that this should be assumed when completing 
, but it may be worthwhile to consider side lengths of the faces touching each other.