query: If $a, b, c$ are reals prove that $ (a^3+b^3) \le (a^2+b^2)•(a^4+b^4) $
positive_passages.docid: 66261
positive_passages.text: Inequalities help: $(a^3+b^3)^2\leq (a^2+b^2)(a^4+b^4)$
negative_passages.docid: 45090
negative_passages.text: Given any positive real numbers $a,b,c$, we have $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
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: 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: 11127
negative_passages.text: Let $a$, $b$, and $c$ be positive real numbers.
negative_passages.docid: 111847
negative_passages.text: For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$
negative_passages.docid: 25258
negative_passages.text: Prove that if $ a,b,c > 0 $, then $ [(1 + a) (1 + b) (1 + c)]^{7} > 7^{7} (a^{4} b^{4} c^{4}) $.
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: 133428
negative_passages.text: How prove this inequality $\frac{a}{b+3}+\frac{b}{c+3}+\frac{c}{d+3}+\frac{d}{a+3}\le 1$
negative_passages.docid: 130281
negative_passages.text: Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
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: 29342
negative_passages.text: Prove the inequality: $(y+z)*x^4+(x+z)*y^4+(x+y)*z^4\le((x+y+z)^5)/12$ for positive reals.
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: 65297
negative_passages.text: Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2} \right |\leq \left |b-c \right |$.
negative_passages.docid: 108801
negative_passages.text: Given that $xyz=1$, prove that $\frac{x}{1+x^4}+\frac{y}{1+y^4}+\frac{z}{1+z^4}\le \frac{3}{2}$
negative_passages.docid: 89838
negative_passages.text: Prove that $3 \le a+b+c \le 2\sqrt{3}$ in a triangle
negative_passages.docid: 68781
negative_passages.text: proving if $a \le b$ then $\sup(A) \le \inf(B)$