1990 - 39, 40, & 41

What is the "AHA!" moment?

Difficulty: Hard.

Now it's time to just make shit up.

Difficulty: Medium - Hard.

Difficulty: Hard.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 1990

answer 39, answer 40, answer 41, Published 5/25.