Powers and roots

Can you find the solution to this algebraic inequality?

What are the last two digits of 2^(2^2003)?

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

Evaluate without a calculator: (5 sqrt2 + 7)^{1/3} - (5 sqrt2 - 7)^1/3}.

What is the last digit of the number 1 / 5^903 ?

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

What is the largest number you can make using the three digits 2, 3 and 4 in any way you like, using any operations you like? You can only use each digit once.

Find the sum of this series of surds.

Can you produce convincing arguments that a selection of statements about numbers are true?

Find the five distinct digits N, R, I, C and H in the following nomogram