parmenides51 wrote: Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{a^3 + b^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$ Let a,b,c>0 with $abc \ge 1 $.Show that: $$\frac{1}{a^2+b^3+c^4}+\frac{1}{b^2+c^3+a^4}+\frac{1}{c^2+a^3+b^4} \le 1$$h

parmenides51 wrote: Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{a^3 + b^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$ any ideas?

Since $0 < a, b, c < 1$, $\sum_{cyc}{\frac{a^2}{b^3 + c^4 + 1}} > \sum_{cyc}{\frac{a^2}{b + c + 1}} \geq \sum_{cyc}{\frac{(a + b + c)^2}{2(a + b + c) + 3}} = \frac{1}{5}$ by Titu's Lemma.

Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that$$\frac{a}{a+b^2+ c^2}+\frac{b}{b+c^2 + a^2}+\frac{c}{c+a^2 + b^2} \leq \frac{9}{5}$$$$\frac{a^2}{a^2+b+ c}+\frac{b^2}{b^2+c+ a}+\frac{c^2}{c^2+a+ b} \geq \frac{3}{7}$$$$\frac{a^2}{a+b^2+ c^2}+\frac{b^2}{b+c^2 + a^2}+\frac{c^2}{c+a^2 + b^2} \geq \frac{3}{5}$$$$ \frac{a^3}{a+b^2+ c^2}+\frac{b^3}{b+c^2 + a^2}+\frac{c^3}{c+a^2 + b^2}\geq \frac{1}{5}$$$$ \frac{a^3}{a^2+b+ c}+\frac{b^3}{b^2+c+ a}+\frac{c^3}{c^2+a+ b}\geq \frac{1}{7}$$$$\frac{a^3}{a^3+b+ c}+\frac{b^3}{b^3+c+ a}+\frac{c^3}{c^3+a+ b}\geq \frac{3}{19}$$

Note that the original inequality isn't a cyclic sum, so #4 is incorrect (or the OP might be incorrect).

I had a huge typo , my first post was parmenides51 wrote: Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{a^3 + b^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$ I had mistyped the second fraction, the correct is Quote: Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{c^3 + a^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$