How do I improve at these types of questions?

Question

Find the smallest cosntant $k>0$ such that $\frac{ab}{a+b+2c} + \frac{bc}{b+c+2a} + \frac{ca}{c+a+2b} \leq k(a+b+c)$ for every $a,b,c>0$.

Basically, questions that involve solving or making sense of equations with like 5 variables and no context. There's a lot of those kinds of questions on contest maths, but I cannot for the life of me solve them. Is there a nice way to approach them or are they all different?

Answer 1

Consider the function $$y= \frac{abc}{x({a+b+c}+x)}$$

Clearly the function is decreasing for positive x and is concave upwards.(You might want to apply first and second derivative test, if you want)

click this image(another 'merit' of my reputation)

The centroid the triangle(brown dot in my diagram) lies always above the curve. I am sure you can take it from there.