Interesting Maths question

I have been interested in maths since I was a child. Sometimes anonymous maths question came to brush my mind. So I think it's nice to share them in my blog. Down here are some lists of interesting maths questions that had me spending some times to think of.

1) If a rectangle has sides of length x and y, the perimeter would be 2x + 2y and area xy. What is the ratio of the two sides so that the rectangle has the smallest perimeter for the biggest area? And what would be the opposite(biggest perimeter for smallest area)?

2) A coin labelled A of diameter 2x is fixed. Another coin labelled B of diameter x is rotated once around the coin A. How many rotation does coin B made? What would be the opposite(B fixed, A rotated around B)?

3) A book and a bookmark cost $5.50. The book is $5.00 more than the bookmark. How much does the bookmark cost?

4) Prove or disprove the equation xⁿ + yⁿ = zⁿ for n< -1, and x,y,z are different integer. (This is a bit different from Fermat's last theorem)

5) A circle has no vertices or infinite vertices?