Giza-IQ Test - Questions 3 and 4

Here are two more questions to test your intelligence with mathematical and geometrical problems.  The first problem I propose to you today was one of the problems I encountered when in 1967 I attended in Rome the selections for the Italian National Math championship.  I qualified for the national championship and there I reached the fifth place, thanks to which I became part of the Italian team sent to the IX International Mathematics Olympiad.

1.03     A string is looped around the tip of a cone and pulled down with a weight, so that it hangs on the cone as shown in the figure below.

The aperture of the cone is indicated in the figure with the greek letter α. Assuming that the contact between the string and the surface of the cone is frictionless, what is the minimum value of α in degrees for which the loop slips off the cone?

1.04     In a regular tetrahedron with unitary edge length, one (and only one) edge is increased in length, thereby increasing the volume of the solid. What is the maximum volume that can be obtained in this way?