query: If $a$, $b$ and $c$ are sides of a triangle, then prove that $a^\text{2}(b+c-a) + b^\text{2}(c+a-b) + c^\text{2}(a+b-c)$ $\leqslant$ $3abc$
positive_passages.docid: 23080
positive_passages.text: Prove that $a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc.$
negative_passages.docid: 68986
negative_passages.text: Use the triangle inequality to show that $|a|+|b| \leq |a+b|+|a-b|$
negative_passages.docid: 52204
negative_passages.text: Proof of one inequality $a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$
negative_passages.docid: 55975
negative_passages.text: Let $a$, $b$ and $c$ be the three sides of a triangle. Show that $\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3$.
negative_passages.docid: 216
negative_passages.text: In a $\triangle ABC,a^2+b^2+c^2=ac+ab\sqrt3$,then the triangle is
negative_passages.docid: 140299
negative_passages.text: Inequality based on triangle: $\frac{3}{2}\le\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$
negative_passages.docid: 61343
negative_passages.text: Prove that if $\alpha, \beta, \gamma$ are angles in triangle, then $(tan(\frac{\alpha}{2}))^2+(tan(\frac{\beta}{2}))^2+(tan(\frac{\gamma}{2}))^2\geq1$
negative_passages.docid: 53660
negative_passages.text: Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q  \implies  \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $
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: 90916
negative_passages.text: Prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$.
negative_passages.docid: 70311
negative_passages.text: Trying to prove (A ∩ B) − C = (A − C) ∩ (B − C)
negative_passages.docid: 138894
negative_passages.text: Prove that if $3\mid(a^2+b^2)$,then $3\mid a$ and $ 3\mid b$
negative_passages.docid: 151038
negative_passages.text: How to prove (A v B), (A → C), (B → D) therefore (C v D)
negative_passages.docid: 148744
negative_passages.text: Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$
negative_passages.docid: 50417
negative_passages.text: Prove that the triangles $ABC$ and $AB^{'}C^{'}$ have the same incentre.
negative_passages.docid: 24499
negative_passages.text: Finding $(a, b, c)$ with $ab-c$, $bc-a$, and $ca-b$ being powers of $2$
negative_passages.docid: 63958
negative_passages.text: how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$