query: If a,b,c are positive real numbers such that a + b + c = 1 , then find the minimum value of $\frac{(1+a)(1+c)(1+c)}{(1-a)(1-b)(1-c)}$
positive_passages.docid: 144164
positive_passages.text: If $x,y,z>0$ are distinct and $x+y+z=1$ what is the minimum of $\left((1+x)(1+y)(1+z)\right)/\left((1-x)(1-y)(1-z)\right)$?
negative_passages.docid: 6970
negative_passages.text: If $abc=1$ and $a,b,c$ are positive real numbers, prove that ${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1$.
negative_passages.docid: 18530
negative_passages.text: Given 4 integers, $a, b, c, d > 0$, does $\frac{a}{b} < \frac{c}{d}$ imply $\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}$?
negative_passages.docid: 87907
negative_passages.text: How to find the maximum and minimum value of $\left|(z_1-z_2)^2 + (z_2-z_3)^2 + (z_3-z_1)^2\right|$ (where $|z_1|=|z_2|=|z_3|=1$)?
negative_passages.docid: 65280
negative_passages.text: If $a, b, c >0$ prove that $ [(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4 $.
negative_passages.docid: 134159
negative_passages.text: Smallest value of $a^2 + b^2 + c^2+ d^2$, given values for $(a+b)(c+d)$, $(a+c)(b+d)$, and $(a+d)(b+c)$
negative_passages.docid: 56101
negative_passages.text: Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$
negative_passages.docid: 61302
negative_passages.text: Find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$
negative_passages.docid: 71299
negative_passages.text: Prove that $\frac{a}{b} < \frac{a+c}{b+d} <\frac{ c}{d}$
negative_passages.docid: 73716
negative_passages.text: Let $x,y,z>0$ and $x+y+z=1$, then find the least value of ${{x}\over {2-x}}+{{y}\over {2-y}}+{{z}\over {2-z}}$
negative_passages.docid: 82317
negative_passages.text: Find the minimum value of $A=\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{2-c^3}{c}$
negative_passages.docid: 111847
negative_passages.text: For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$
negative_passages.docid: 63071
negative_passages.text: the least possible value for :$ \lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor $
negative_passages.docid: 40700
negative_passages.text: Proving inequality $(a+\frac{1}{a})^2 + (b+\frac{1}{b})^2 \geq \frac{25}{2}$ for $a+b=1$
negative_passages.docid: 15116
negative_passages.text: Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$
negative_passages.docid: 142076
negative_passages.text: $c$ is a complex number that satisyfing $(c+\frac{1}{c}+1)(c+\frac{1}{c}) = 1$
negative_passages.docid: 90670
negative_passages.text: Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.