In recent times, I realize that asking good questions is just as important as answer them. I mean, we, mathematicians are not just answer-machines, and mathematics is, overall, a human activity.
But there is -in academia- a very intense enphasis on answering questions and so little enphasis on making good questions. Is there a way to develop such an ability? Maybe even a book?
I think that this quote attributed to Riemann could help you:
If only I had the Theorems! Then I should find the proofs easily enough.
This quote goes at the heart of your question and it firmly guards your attitude about this topic.
Sometimes good math questions came from similar questions by generalization, by strengthening,...
For example.
There is known Nesbitt's inequality:
For positives $a$, $b$ and $c$ prove that: $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}.$$
Shapiro created the following.
For all positives $x_i$ prove that $$\frac{x_1}{x_2+x_3}+\frac{x_2}{x_3+x_4}+...+\frac{x_n}{x_1+x_2}\geq\frac{n}{2},$$ which is not true for all $n$, but has beautiful proofs for $n\in\{3,4,5,6\}$ and has very interesting history.
Also, we can make the following.
Since $$a+b\leq\sqrt{2(a^2+b^2)}\leq\sqrt[3]{4(a^3+b^3)}\leq...,$$ we can try to make something stronger: $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{\sqrt{2(a^2+b^2)}}\geq\frac{3}{2};$$ $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{\sqrt[3]{4(a^3+b^3)}}\geq\frac{3}{2};$$ $$.$$ $$.$$ $$.$$ We obtain here that $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{\sqrt[5]{16(a^5+b^5)}}\geq\frac{3}{2}$$ is wrong.
Good luck!
