I have a lot of interests in mathematics. Over the course of 10 years, I have learnt in my opinion, a fair share of knowledges and mathematics. Just until recently I encountered several problems that look like this:
Let $a_1, a_2, a_3,...,a_n$ be a sequence of positive numbers. Show that for all positive x, \begin{equation} (x+a_1)(x+a_2) ...(x+a_n) \leq \left(x + \frac{a_1 + a_2 +...+a_n}{n}\right)^n \end{equation}
I have no idea how to solve this, even after trying for a long time, try to gather resources which seems to related to this problems. I can't think of the relations between those data and books, or sources that I gathered, to the thing I am desperately need answer to, or a hint, until I open the solution manual which contains the solution. I was fascinated that it is so simple and can be easily understood, a beautiful, elegant solution to the problem. But I can't figure it out myself.
That is when I realized that I have no logical knowledge about how mathematical inequalities work. I have no intuition, no "mindset" to even try to investigate the right way. A lot of resources, which for example, "Secret of Inequalities" by Pham Kim Hung, eventually is too far of from what I want as a beginner friendly introduction to the field.
What I want to humbly ask is just very simple, and I think a lot of people out there also have similar question and problem like me:
- A comprehensive resource (textbook or websites, online, etc) that literally go from the total beginning. Not from arithmetic, but assumed you have the basic understanding of algebra to go further and deeper. No "slam a bunch of theorem and equation" and then hard example, but to discover the world of inequalities by heart and and by the intriguing mind of human.
- A "surplus" amount of exercises. When I started learning about calculus, eventually from calculus I to III, I already knew that math can't be learn without practice, just like anything else. You can't know how to write simply by reading other's writing ang guideline, or know how to play football without even touch the ball. This is what I think a lot of people want, and also me: A numerous, massive stockpile of questions and problems, from the simplest then gradually to the hardest.
Are there any resouces like this? Where can I find them?