Found problems: 61
Let $k$ and $m$ be positive integers. Show that \[S(m, k)=\sum_{n=1}^{\infty}\frac{1}{n(mn+k)}\] is rational if and only if $m$ divides $k$.
If $x$ is a positive rational number show that $x$ can be uniquely expressed in the form $x = \sum^n_{k=1} \frac{a_k}{k!}$ where $a_1, a_2, \ldots$ are integers, $0 \leq a_n \leq n - 1$, for $n > 1,$ and the series terminates. Show that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^6.$
Let $ \alpha$ be a rational number with $ 0 < \alpha < 1$ and $ \cos (3 \pi \alpha) \plus{} 2\cos(2 \pi \alpha) \equal{} 0$. Prove that $ \alpha \equal{} \frac {2}{3}$.
Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $ab$ and $a+b$, and having the property that for every rational number $r$ exactly one of the following three statements is true:
$$r\in S,\;\; -r\in S,\;\;r =0.$$
Prove that $S$ is the set of all positive rational numbers.
Show that there are infinitely many rational triples $(a, b, c)$ such that $$a + b + c = abc = 6.$$
Prove that on the coordinate plane it is impossible to draw a closed broken line such that
[i](i)[/i] the coordinates of each vertex are rational;
[i](ii)[/i] the length each of its edges is 1;
[i](iii)[/i] the line has an odd number of vertices.
Prove that every positive rational number can be represented in the form \[\frac{a^{3}+b^{3}}{c^{3}+d^{3}}\] for some positive integers $a, b, c$, and $d$.
Prove that there are no rational numbers $x,y,z$ with $x+y+z=0$ and $x^2+y^2+z^2=100$.
Find all $x$ and $y$ which are rational multiples of $\pi$ with $0<x<y<\frac{\pi}{2}$ and $\tan x+\tan y =2$.
Which fractions $ \dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\sqrt{2}$ (without using any table).
Let $W(x)$ be a polynomial of integer coefficients such that for any pair of different rational number $r_1$, $r_2$ dependence $W(r_1) \neq W(r_2)$ is true. Decide, whether the assuptions imply that for any pair of different real numbers $t_1$, $t_2$ dependence $W(t_1) \neq W(t_2)$ is true.
Let $S=\{x_0, x_1, \cdots, x_n\} \subset [0,1]$ be a finite set of real numbers with $x_{0}=0$ and $x_{1}=1$, such that every distance between pairs of elements occurs at least twice, except for the distance $1$. Prove that all of the $x_i$ are rational.
Let $a$ and $b$ be some rational numbers and there exist $n$, such that $\sqrt[n]{a}+\sqrt[b]{b}$ is also a rational number. Prove that $\sqrt[n]{a}$ is a rational number.
Let $f(t)=t^3+t$. Decide if there exist rational numbers $x, y$ and positive integers $m, n$ such that $xy=3$ and:
\begin{align*}
\underbrace{f(f(\ldots f(f}_{m \ times}(x))\ldots)) = \underbrace{f(f(\ldots f(f}_{n \ times}(y))\ldots)).
\end{align*}
Find all rational numbers $k$ such that $0 \le k \le \frac{1}{2}$ and $\cos k \pi$ is rational.
Which fractions $ \dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\sqrt{2}$ (without using any table).
The infinity sequence $r_{1},r_{2},...$ of rational numbers it satisfies that:
$\prod_{i=1}^ {k}r_{i}=\sum_{i=1}^{k} r_{i}$.
For all natural k.
Show that $\frac{1}{r_{n}}-\frac{3}{4}$ is a square of rationale number for all natural $n\geq3$
Let $ \alpha$ be a rational number with $ 0 < \alpha < 1$ and $ \cos (3 \pi \alpha) \plus{} 2\cos(2 \pi \alpha) \equal{} 0$. Prove that $ \alpha \equal{} \frac {2}{3}$.
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
Is it possible to find $p,q,r\in\mathbb Q$ such that $p+q+r=0$ and $pqr=1$?
[i]Proposed by Máté Weisz, Cambridge[/i]
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
Suppose that $\tan \alpha =\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove the number $\tan \beta$ for which $\tan 2\beta =\tan 3\alpha$ is rational only when $p^2 +q^2$ is the square of an integer.
The numbers $1- \sqrt{2}$, $\sqrt{2}$ and $1+\sqrt{2}$ are written on a blackboard. Every minute, if $x, y, z$ are the numbers written, then they are erased and the numbers, $x^2 + xy + y^2$, $y^2 + yz + z^2$ and $z^2 + zx + x^2$ are written. Determine whether it is possible for all written numbers to be rational numbers after a finite number of minutes.
$a, b$ and $c$ are positive integers and \[\frac{a\sqrt{3} + b}{b\sqrt{3} + c}\]
is a rational number.
Show that \[\frac{a^2 + b^2 + c^2}{a + b + c}\] is an integer.