<s>[INST] Prove that $3(a^3+b^3+c^3) > (a+b+c)(a^2+b^2+c^2)$ Let $a,b,c$ be positive, not equal. Prove that $3(a^3+b^3+c^3) > (a+b+c)(a^2+b^2+c^2)$.
I know the proof by subtracting LHS by RHS and then doing some arrangement.
But isn't there any inequality which can be used in $(1+1+1)(a^3+b^3+c^3) >(a+b+c)(a^2+b^2+c^2)$, or an easy proof?
 [/INST] There is a fairly general way to prove inequalities like this using rearrangement, but just for fun, let me show you how you can prove a large class of inequalities like this by repeated use of AM-GM. First subtract $a^3, b^3, c^3$ from the RHS, then symmetrically sum the inequalities
$$a^3 + a^3 + b^3 \ge 3 a^2 b.$$
Essentially I am using weighted AM-GM with rational coefficients. Here are some exercises that can be solved using this idea.
Exercise 1. Prove that if $n > 1$ is a positive integer and $a > 0$, then 
$$\frac{1 + a + ... + a^n}{a + a^2 + ... + a^{n-1}} \ge \frac{n+1}{n-1}.$$
Exercise 2. Let $a, b, c \ge 0$. Prove that 
$$a^5 + b^5 + c^5 \ge 5abc(b^2 - ac).$$
(Weighted AM-GM isn't necessarily the best way to solve either of these problems; it's just the argument I came up with first.) 
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