query: inequality proof $3*(a^3+b^3+c^3) > (a+b+c)*(a^2+b^2+c^2)$
positive_passages.docid: 86378
positive_passages.text: Prove that $3(a^3+b^3+c^3) > (a+b+c)(a^2+b^2+c^2)$
negative_passages.docid: 29462
negative_passages.text: Suggestions on how to prove the following equality. $a^{m+n}=a^m a^n$
negative_passages.docid: 80789
negative_passages.text: Prove that $\forall a,b,c,d\in \mathbb{R}\,$, $(a^2-b^2)(c^2-d^2)\leq(ac-bd)^2$ and equalities hold if and only if $ad=bc$.
negative_passages.docid: 37536
negative_passages.text: Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$.
negative_passages.docid: 85763
negative_passages.text: Inequality $(1+\frac1k)^k \leq 3$
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: 104498
negative_passages.text: Show that $x^3y^3(x^3+y^3) \leq 2$
negative_passages.docid: 109133
negative_passages.text: $a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$
negative_passages.docid: 138216
negative_passages.text: $a,b,c$ are positive real numbers such that, $a+b+c\ge abc$. Prove that $a^2+b^2+c^2\ge \sqrt{3}abc$
negative_passages.docid: 117281
negative_passages.text: How prove this inequality $x^3y+y^3z+z^3x\ge xyz(x+y+z)$
negative_passages.docid: 63958
negative_passages.text: how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$
negative_passages.docid: 154716
negative_passages.text: Prove $\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+c^3+abc}+\frac{1}{b^3+c^3+abc} \leq \frac{1}{abc}$
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: 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: 45619
negative_passages.text: Prove: $|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$
negative_passages.docid: 78144
negative_passages.text: How to prove the inequality $a_1^{a_1}a_2^{a_2}\cdots a_n^{a_n} > \left(\frac{a_1+\dots+a_n}{n}\right)^{a_1+\dots+a_n}$?
negative_passages.docid: 141525
negative_passages.text: Cyclic Inequality $\frac{a^3}{(a+b)(a+c)}$