query: If $x, y \in \mathbb{N^{*}}$, such that $\frac{x}{y}+\frac{y}{x} \in \mathbb{N}$, show that $x=y$.
positive_passages.docid: 62792
positive_passages.text: Can sum of a rational number and its reciprocal be an integer?
negative_passages.docid: 144473
negative_passages.text: Finding all functions $ f : \mathbb N^*\to\mathbb N^*$ such that for every $ x , y \in \mathbb N ^ * $, $ f ( x ) + f ( y ) \mid x ^ n + y ^ n $
negative_passages.docid: 108429
negative_passages.text: Show that if $x_n \to x$ then $\sqrt{x_n} \to \sqrt{x}$
negative_passages.docid: 153355
negative_passages.text: Given $S \in B(Y^{*}, X^{*})$, does there exist $T\in B(X,Y)$ such that $S=T^{*}$?
negative_passages.docid: 21891
negative_passages.text: If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t
negative_passages.docid: 34693
negative_passages.text: Prove $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$ if $E^{\mathbb P}[X]=\mathbb P(X>0)=1$ and $ \mathbb Q(A)=E^{\mathbb Q}[X1_A] $
negative_passages.docid: 63553
negative_passages.text: Show that $\frac{1}{(n+1)!}(1+\frac{1}{(n+1)}+\frac{1}{(n+1)^2}+\frac{1}{(n+1)^3}+\cdots)=\frac{1}{n!n}$
negative_passages.docid: 1932
negative_passages.text: How to show this equality of $\rho(X,Y)$
negative_passages.docid: 99933
negative_passages.text: Determine whether $(x_n)$ converges if $x_1= a , x_{n+1} = x_n+\frac{1}{x_n}$ for $n \in \mathbb{N}$
negative_passages.docid: 133451
negative_passages.text: To show that ${x_n}$ is convergent where $|x_{n+1} - x_n|< \frac{1}{n^2}$
negative_passages.docid: 63183
negative_passages.text: Suppose $a,b,m,n$ are natural numbers, prove that if $a({m \over n})=b$ and $(a,n)=1$, then ${m \over n} \in \mathbb{N}$
negative_passages.docid: 62508
negative_passages.text: If $x$, $y$, $z$ are in arithmetic progression, show that $\frac{\sin x + \sin y + \sin z }{\cos x + \cos y + \cos z} = \tan y. $
negative_passages.docid: 14197
negative_passages.text: show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
negative_passages.docid: 32407
negative_passages.text: Prove $\exp(x+y) = \exp(x) \exp(y)$ for $\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}$
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: 71299
negative_passages.text: Prove that $\frac{a}{b} < \frac{a+c}{b+d} <\frac{ c}{d}$
negative_passages.docid: 121455
negative_passages.text: $f(x)$ if $f(xy)=f(x) +f(y) +\frac{x+y-1}{xy}$