In this note, I have a selection of questions for you to prove. It is for all levels to try. The theme is algebra. Good luck.
1) Prove that
for any positive real numbers , and .
2) Find all functions such that
for all real numbers and .
3) Consider the polynomial
(a) Prove that the equation has exactly three real solutions.
(b) If the solutions to are , and , find a monic cubic polynomial whose roots are , and .
4) The equation has four real positive roots.
5) A sequence of real functions is defined by
Find all real solutions of the equation
6) Let be a positive integer. Suppose that are positive real numbers such that