query: $x + \frac1y = y + \frac1z = z + \frac1x$, then value of $xyz$ is?
positive_passages.docid: 20957
positive_passages.text: If $a+\frac1b=b+\frac1c=c+\frac1a$ for distinct $a$, $b$, $c$, how to find the value of $abc$?
negative_passages.docid: 62758
negative_passages.text: How find this value $\frac{a^3}{(a-p)(a-q)}+\frac{b^3}{(b-p)(b-q)}+\frac{c^3}{(c-p)(c-q)}$
negative_passages.docid: 61186
negative_passages.text: For which $n$ does $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ imply $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$
negative_passages.docid: 16774
negative_passages.text: If $x,y,z\in\mathbb R\setminus \{1\}$ and $xyz=1$, prove that $\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1$.
negative_passages.docid: 136586
negative_passages.text: Making $x+\frac1y$, $y+\frac1z$, and $z+\frac1x$ all integer.
negative_passages.docid: 90786
negative_passages.text: What are $x$, $y$ and $z$ if $\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y} = 4$ and $x$, $y$ and $z$ are whole numbers?
negative_passages.docid: 72906
negative_passages.text: Is $\frac{\partial}{\partial x_1}=\frac{\partial}{\partial y_1}$ if $x_1=y_1$?
negative_passages.docid: 144644
negative_passages.text: Solution of $\frac{dy}{dx}=\frac{1}{xy(x^2 \sin y^2+1)}$
negative_passages.docid: 96265
negative_passages.text: Why does $x= \frac{1}{2}(z+\bar{z}) = \frac{1}{2}(z+\frac{r^{2}}{z})$ on the circle?
negative_passages.docid: 10444
negative_passages.text: If $\{x,\{x,y\}\} = \{z, \{z,t\}\}$, then must it be $x = z$ and $y = t$?
negative_passages.docid: 113334
negative_passages.text: If $x+\frac{1}{x}=\frac{1+\sqrt{5}}{2}$ then $x^{2000}+\frac{1}{x^{2000}}= $?
negative_passages.docid: 98653
negative_passages.text: Easiest way to solve $y''+y=\frac{1}{\cos x}$
negative_passages.docid: 45319
negative_passages.text: Properties of $y$ if $\frac{d^2y}{dx^2}+xy=0$, $x \in [a,b],$ and $y(a)=y(b)=0.$
negative_passages.docid: 22528
negative_passages.text: maximum value of $xy+yz+zx.$ given $x+2y+z=4$
negative_passages.docid: 4278
negative_passages.text: Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true?
negative_passages.docid: 14410
negative_passages.text: If $(1+ \cos A) (1+ \cos B) (1+\cos C)= y = (1- \cos A) (1-\cos B) (1-\cos C)$ then $y = \pm \sin A \sin B \sin C$
negative_passages.docid: 142076
negative_passages.text: $c$ is a complex number that satisyfing $(c+\frac{1}{c}+1)(c+\frac{1}{c}) = 1$