query: If $xy$ divides $x^2 + y^2$ show that $x=\pm y$
positive_passages.docid: 156646
positive_passages.text: If $x$ is rational then $\,x+\frac{1}x \,$ is an integer $\iff x = \pm 1$
negative_passages.docid: 53725
negative_passages.text: Factor $(x+y)^4+x^4+y^4$
negative_passages.docid: 73073
negative_passages.text: Differentiating $y=x^{2}$
negative_passages.docid: 152643
negative_passages.text: $(xy + 1)(yz + 1)(zx + 1) $ is a perfect square if and only if each factor is
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: 41383
negative_passages.text: Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$
negative_passages.docid: 120736
negative_passages.text: If xy + yz + zx = 1, ...........
negative_passages.docid: 18313
negative_passages.text: show this inequality with $xy+yz+zx=3$
negative_passages.docid: 87029
negative_passages.text: Solve $y^2=x^3-4$ in $x,y\in \mathbb{Z}$
negative_passages.docid: 138931
negative_passages.text: Show that $||x+y||\cdot||x-y||\le||x||^2+||y||^2$
negative_passages.docid: 86594
negative_passages.text: If $x^2 \equiv y^2 \pmod {p^r}$, where $p$ is an odd prime not dividing $x$ or $y$ then $x \equiv \pm y \pmod {p^r}$
negative_passages.docid: 20467
negative_passages.text: Show that $(y_n)\in l^q$
negative_passages.docid: 11693
negative_passages.text: If $p \mid m^p+n^p$ prove $p^2 \mid m^p+n^p$
negative_passages.docid: 127183
negative_passages.text: Proof: For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$
negative_passages.docid: 76640
negative_passages.text: Show that the polynomial $x^3-xy^2+y+1$ is irreducible in $\mathbb{Q}[x,y]$
negative_passages.docid: 48038
negative_passages.text: If $z$ and $w$ are complex numbers such that $|(z+w)| = |(z-w)|$, prove that $\arg(z)-\arg(w) = \pm (\pi/2)$
negative_passages.docid: 21737
negative_passages.text: If $A|B$ and $B|A$ then prove $A=\pm B$