query: Let $a,b,c$ be the length of sides of a triangle then prove that $a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge0$
positive_passages.docid: 45702
positive_passages.text: For any triangle with sides $a$, $b$, $c$, show that $a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0$
negative_passages.docid: 115929
negative_passages.text: In a triangle $\angle A = 2\angle B$ iff $a^2 = b(b+c)$
negative_passages.docid: 5694
negative_passages.text: How to prove $a^2 + b^2 + c^2 \ge ab + bc + ca$?
negative_passages.docid: 70311
negative_passages.text: Trying to prove (A ∩ B) − C = (A − C) ∩ (B − C)
negative_passages.docid: 50296
negative_passages.text: Prove $(-a+b+c)(a-b+c)(a+b-c) \leq abc$, where $a, b$ and $c$ are positive real numbers
negative_passages.docid: 139082
negative_passages.text: If $a,b,c>0$ and $a^3+b^3=c^3$ then prove that $a^2+b^2-c^2>6(c-a)(c-b)$.
negative_passages.docid: 113380
negative_passages.text: Prove that $\left | \frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a} \right | < \frac{1}{8}.$
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: 121177
negative_passages.text: Prove that $\frac{(2a+b+c)^2}{2a^2+(b+c)^2}+\frac{(2b+c+a)^2}{2b^2+(c+a)^2}+\frac{(2c+a+b)^2}{2c^2+(a+b)^2} \le 8$
negative_passages.docid: 118507
negative_passages.text: Prove $\frac{ab}{1+c^2}+\frac{bc}{1+a^2}+\frac{ca}{1+b^2}\le\frac{3}{4}$ if $a^2+b^2+c^2=1$
negative_passages.docid: 141102
negative_passages.text: If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$.
negative_passages.docid: 6621
negative_passages.text: Prove the inequality $a^2bc+b^2cd+c^2da+d^2ab \leq 4$ with $a+b+c+d=4$
negative_passages.docid: 73029
negative_passages.text: If a,b,c are sides of a triangle, prove: $ \sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \le \sqrt{a} + \sqrt{b} + \sqrt{c} $
negative_passages.docid: 124868
negative_passages.text: Prove that $\frac{bc}{b+c}+\frac{ca}{c+a}+\frac{ab}{a+b}<\frac{a+b+c}{2}$
negative_passages.docid: 137791
negative_passages.text: Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$
negative_passages.docid: 54170
negative_passages.text: Prove that $\triangle ABC$ is equilateral, If $(a+b-c)^{a}\cdot (b+c-a)^{b}\cdot (c+a-b)^{c}\geq a^a\cdot b^b\cdot c^c$
negative_passages.docid: 111847
negative_passages.text: For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$