Inequalities

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?

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?

Can you find the solution to this algebraic inequality?

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.