This is about Math Challenge III Algebra.

Problem 4.17: do something similar to 4.8.

Problem 4.19: Try to apply Cauchy-Schwarz.

Problem 4.20: Note that $x^2+y^2$ is related to a circle.  Some change of variables will be helpful.

Problem 4.22 is a straightforward problem with absolute values; you need to do case work.

Problem 4.23: Take base 4 log.  This is pretty standard as well.

Problem 4.24: Logarithm is usually convertible to the corresponding exponents.  You have to consider the cases $0 < x < 1$ and $x > 1$.

Problem 4.25: Similar to 4.17 and 4.8, but need some more clever handling.

Problem 4.26: application of the common inequalities such as AM-GM.

Problem 4.28 and 4.29: change of variables will help.

Problem 4.30: If you consider the two square roots as two distance formulas ...

Let me just go this far.  The techniques involved in these questions are really helpful.  Note that some questions are indeed pretty hard, so don't worry if you can't solve them.  We will go over the methods so you can learn and practice to use them.