Inequality to prove Please, I need to prove this for positive reals $a,b,c$: $a^3+b^3+c^3 \geq a^2b+b^2c+c^2a$ I know many things involving maths(I am graduating in half-year time) but, here I am an absolute beginner, so, please be as 'methodic' as you can.

The most elementary poof I know goes as follows:

By AM-GM we have $\frac{a^3+a^3+b^3}{3}≥\sqrt[3]{a^3a^3b^3}=a^2b$ if you do this cyclicly for every variable and add the inequalities, you obtain the inequality to prove.

Its kind of hard to give a universal method to solve such inequalities, this is why its a topic in many mathematical olympiads. However, a good thing to do in order to learn solving them is to understand and to use the elementary inequalities such as the mean inequalities, Cauchy-Schwarz, Chebychev, Arrangement-inequality and so on.