query: For positive, real $a$, $b$, $c$ with $a+b+c=1$, show $\sum_{cyc}\frac{a^3}{a^2+b^2}\geq \frac12$
positive_passages.docid: 99820
positive_passages.text: symetric inequality for a rational function of three variables
negative_passages.docid: 101362
negative_passages.text: Let a,b,c be positive real number, proof.
negative_passages.docid: 107494
negative_passages.text: prove $\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_i}{a_i+b_i}$
negative_passages.docid: 99676
negative_passages.text: Prove $\sum{\frac{1}{(x+2y)^2}} \geq\frac{1}{xy+yz+zx}$
negative_passages.docid: 28499
negative_passages.text: Inequality.$\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq 3$
negative_passages.docid: 90670
negative_passages.text: Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.
negative_passages.docid: 134535
negative_passages.text: Inequality $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a + b+ c$ when $abc = 1$
negative_passages.docid: 112565
negative_passages.text: Prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{9}{a+b+c} : (a, b, c) > 0$
negative_passages.docid: 60673
negative_passages.text: To prove $\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$
negative_passages.docid: 152053
negative_passages.text: Proof that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a+b+c$
negative_passages.docid: 90902
negative_passages.text: Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$.
negative_passages.docid: 44247
negative_passages.text: For $a^3+b^3+c^3=3$ prove that $a^4b+b^4c+c^4a\leq3$
negative_passages.docid: 137864
negative_passages.text: Prove that $ \left(1+\frac a b \right) \left(1+\frac b c \right)\left(1+\frac c a \right) \geq 2\left(1+ \frac{a+b+c}{\sqrt[3]{abc}}\right)$.
negative_passages.docid: 157416
negative_passages.text: Prove inequality: $\sum \frac{a^4}{a^3+b^3} \ge \frac{a+b+c}{2}$
negative_passages.docid: 79491
negative_passages.text: Inequality $(a+b)^2 + (a+b+4c)^2\ge \frac{kabc}{a+b+c}$ for $a,b,c \in\mathbb{R}$
negative_passages.docid: 104498
negative_passages.text: Show that $x^3y^3(x^3+y^3) \leq 2$
negative_passages.docid: 82612
negative_passages.text: Show that $\frac c {1+c} \le \frac a {1+a} + \frac b {1+b}$ , for $c \le a+b$ and $a,b,c \ge 0$