Mathematical puzzles

Some of dreams still follow logic in this case I was present at a quiz evening in which the following questions were asked and I was roped into answering them, which I did so while still dreaming. Upon awakening I jotted them all down.

I have no idea where the first question and answer came from; the second is a variant on a different puzzle and the third from a magic book I read aged 12.

Problem 1: An airport owner has 7 airports, each offers a direct flight to the other and the owner wishes a diagram that shows this. Show all the direct flights using only four lines. Oh and drawing a line from airport 1 to airport 3 that passes through airport 2 is not 'one line' it's two, because it doesn't show the direct flight from 1-3.

Problem 2: Take a cube (six sides, square faces) and a tetrahedron (4 sides, triangular faces) and connect a face of the former to the latter so all sides and corners of the faces meet.

Problem 3: Make a hole in an A4 sheet of paper large enough for an adult human to pass through.

Solutions to follow; have fun.