2.$\sum_{cyc} \dfrac{a^5}{a^3+1} =\sum_{cyc}\dfrac{a^4}{a^2+bc} \ge \sum_{cyc}\dfrac{(a^2+b^2+c^2)^2}{ab+bc+ba+a^2+b^2+c^2} \ge \dfrac{1}{2}(a^2+b^2+c^2)\ge \dfrac{3}{2}$
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Hope is what makes us strong.
It's why we are here.
It is what we fight with when all else is lost.
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