<s>[INST] Min of $\frac{a^{10}+ b^{10}}{a^{7}+ b^{7}} +\frac{b^{10}+ c^{10}}{b^{7}+ c^{7}} +\frac{c^{10}+ a^{10}}{c^{7}+ a^{7}} $ I got this problem I tried several time to solve it by many inequalities but I got stuk. My question is how I get the minimum value of  $$ \frac{a^{10}+ b^{10}}{a^{7}+ b^{7}} +\frac{b^{10}+ c^{10}}{b^{7}+ c^{7}} +\frac{c^{10}+ a^{10}}{c^{7}+ a^{7}} $$ if you know that  $a, b, c \in (0, \infty) $ and $ a+b+c=1$?  Any hint? 
 [/INST] $2(a^{10}+b^{10})\ge (a^7+b^7)(a^3+b^3)$ with chebishev inequality, then $$\frac{a^{10}+b^{10}}{a^7+b^7}+\frac{b^{10}+c^{10}}{b^7+c^7}+\frac{a^{10}+c^{10}}{a^7+c^7} \ge a^3+b^3+c^3 \ge 3 (\frac{a+b+c}{3})^3=\frac{1}{9}$$ because with power mean we have $ (\frac{a^3+b^3+c^3}{3})^\frac{1}{3} \ge (\frac{a+b+c}{3})$ 
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