<s>[INST] Prove or disprove inequality $a^2+b^2+c^2\ge a^rb^{2-r}+b^rc^{2-r}+c^ra^{2-r}$. If $a$, $b$ and $c$ are real numbers greater than $0$ and $r$ is a real number with $0 \le r \le 2$. Does inequality $$a^2+b^2+c^2\ge  a^rb^{2-r}+b^rc^{2-r}+c^ra^{2-r}$$ hold?
 [/INST] Hint: Use Rearrangement Inequality, noting $a^r, b^r, c^r$ and $a^{2-r}, b^{2-r}, c^{2-r}$ are similarly ordered.
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