How to make good questions in math? Is there a way to practice this ability? 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?
 A: 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.
A: 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!
