I am interested in finding out more information on references regarding recreational mathematics, specifically when working with inequalities. I enjoy working on problems from http://awesomemath.org under Mathematical Reflections. Many of the problems deal with proving that the following inequality holds, for example:
Let $a,b,c$ be positive real numbers such that $\frac{1}{\sqrt{1+a^3}}+\frac{1}{\sqrt{1+b^3}}+\frac{1}{\sqrt{1+c^3}} \leq 1$. Prove that $a^2+b^2+c^2 \geq 12$ (from Mathematical Reflections -Issue 3 2017).
Upon looking at some of the older solutions, many solutions use the AM-GM inequality or other neat tricks involving rewriting the expression or a substitution.
I do not have a strong background regarding these types of problems. I am hoping to find references that will help me with working on these types of questions-books, articles, similar questions.
Thank you.
Try Putnam and Beyond by Titu Andreescu! http://www-bcf.usc.edu/~lototsky/PiMuEp/PutnamAndBeyond-Andreescu.pdf