query: When is $x^2+y^2+1=xyz$?
positive_passages.docid: 25784
positive_passages.text: Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$.
negative_passages.docid: 136933
negative_passages.text: Which $n$ can be expressed as $x^2 − y^2$?
negative_passages.docid: 111467
negative_passages.text: find all natural solution that satisfy $x^2+y^2 = 3z^2$
negative_passages.docid: 28218
negative_passages.text: Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$
negative_passages.docid: 145287
negative_passages.text: Solve for integers $x, y, z$ such that $x + y = 1 - z$ and $x^3 + y^3 = 1 - z^2$.
negative_passages.docid: 15025
negative_passages.text: Solve $xu_x+(x+y)u_y=1$ when $u(1,y)=y$
negative_passages.docid: 71802
negative_passages.text: $x + \frac1y = y + \frac1z = z + \frac1x$, then value of $xyz$ is?
negative_passages.docid: 59239
negative_passages.text: For non-zero real $x$, $y$, $z$ and complex $a$, $b$, $c$ with $|a|=|b|=|c|$, if $x+y+z=0=ax+by+cz$, then $a=b=c$
negative_passages.docid: 54370
negative_passages.text: If $x+y+z=xyz$, prove $\frac{2x}{1-x^2}+\frac{2y}{1-y^2}+\frac{2z}{1-z^2}=\frac{2x}{1-x^2}\times\frac{2y}{1-y^2}\times\frac{2z}{1-z^2}$
negative_passages.docid: 139930
negative_passages.text: Find all polynomials $p$ such that $p(x^2)=p(x)p(x+1)$
negative_passages.docid: 100278
negative_passages.text: Find $\alpha^{2016} + \beta^{2016} + \alpha^{2014} + \beta^{2014} \over \alpha^{2015} + \beta^{2015}$ for zeroes of the polynomial $x^2+3x +1$
negative_passages.docid: 86211
negative_passages.text: If $x+y+z \geq xyz$ Prove that $x^2 + y^2 + z^2 \geq xyz$.
negative_passages.docid: 12837
negative_passages.text: Inequality with condition $x^2+y^2+z^2=1$.
negative_passages.docid: 41689
negative_passages.text: Find all solutions: $x^2 + 2y^2 = z^2$
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: 135365
negative_passages.text: If $x, y, z$ are the side lengths of a triangle, prove that $x^2 + y^2 + z^2 < 2(xy + yz + xz)$
negative_passages.docid: 155363
negative_passages.text: Find all integers $(x, y)$ such that $1 + 2^x + 2^{2x + 1} = y^2$