Inequalities

When I park my car in Mathstown, there are two car parks to choose from. Can you help me to decide which one to use?

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

Find the maximum value of n to the power 1/n and prove that it is a maximum.

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?

The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?

Is it possible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units?

Which is the bigger, 9^10 or 10^9 ? Which is the bigger, 99^100 or 100^99 ?

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

Can you build a distribution with the maximum theoretical spread?

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?