<s>[INST] If $a,b,c$ are distinct positive numbers , show that $\frac{a^8 + b^8 + c^8}{a^3 b^3 c^3} > \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ If $a,b,c$ are distinct positive numbers , show that $$\frac{a^8 + b^8 + c^8}{a^3 b^3 c^3} > \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$ I am thinking of Tchebycheff's inequality for this question, but not able to proceed. How do I solve this?
 [/INST] Use the inequality $x^2+y^2+z^2>zy+yz+zx$ for distinct $x,y,z$ repeatedly: $$a^8+b^8+c^8>a^4b^4+b^4c^4+c^4a^4>a^4b^2c^2+a^2b^4c^2+a^2b^2c^4>a^3b^3c^2+a^2b^3c^3+a^3b^2c^3.$$
</s>