Found problems: 102
Let $p$ be a prime. Show that $\sqrt[3]{p} +\sqrt[3]{p^5} $ is irrational.
Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.
Find all positive integers $k$ for which there is an irrational $\alpha>1$ and a positive integer $N$ such that $\left\lfloor\alpha^{n}\right\rfloor$ is of the form $m^2-k$ com $m \in \mathbb{Z}$ for every integer $n>N$.
Let the roots of $x^2-3x+1=0$ be $r$ and $s$. Then the expression $r^2+s^2$ is:
$ \textbf{(A)}\ \text{a positive integer} \qquad\textbf{(B)}\ \text{a positive fraction greater than 1}\qquad\textbf{(C)}\ \text{a positive fraction less than 1}$
$\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}$
Find the smallest positive integer $n$ such that \[0< \sqrt[4]{n}-\lfloor \sqrt[4]{n}\rfloor < 0.00001.\]
Suppose $\alpha, \beta$ are two positive rational numbers. Assume for some positive integers $m,n$, it is known that $\alpha^{\frac 1n}+\beta^{\frac 1m}$ is a rational number. Prove that each of $\alpha^{\frac 1n}$ and $\beta^{\frac 1m}$ is a rational number.
We call a positive integer $q$ a $convenient \quad denominator$ for a real number $\alpha$ if
$\displaystyle |\alpha - \dfrac{p}{q}|<\dfrac{1}{10q}$ for some integer $p$. Prove that if two irrational numbers $\alpha$ and
$\beta$ have the same set of convenient denominators then either $\alpha+\beta$ or $\alpha- \beta$ is an integer.
Given a set $S$ of $2n-1$, $n\in \mathbb N$, different irrational numbers. Prove that there are $n$ different elements $x_1, x_2, \ldots, x_n\in S$ such that for all non-negative rational numbers $a_1, a_2, \ldots, a_n$ with $a_1+a_2+\ldots + a_n>0$ we have that $a_1x_1+a_2x_2+\cdots +a_nx_n$ is an irrational number.
Show that $\frac{1}{\pi} \arccos \left( \frac{1}{\sqrt{2003}} \right)$ is irrational.
Let $a_1, a_2, ... , a_{2012}$ be odd positive integers. Prove that the number
$$A=\sqrt{a^2_1+ a^2_2+ ...+ a^2_{2012}-1}$$ is irrational.
Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with a rational area.
Prove that there exist positive integers $ m$ and $ n$ such that
\[ \left\vert\frac{m^{2}}{n^{3}}\minus{}\sqrt{2001}\right\vert <\frac{1}{10^{8}}.\]
Let $1<a_{1}<a_{2}<\cdots$ be a sequence of positive integers. Show that \[\frac{2^{a_{1}}}{{a_{1}}!}+\frac{2^{a_{2}}}{{a_{2}}!}+\frac{2^{a_{3}}}{{a_{3}}!}+\cdots\] is irrational.
Show that $\cos \frac{\pi}{7}$ is irrational.
Suppose $\alpha, \beta$ are two positive rational numbers. Assume for some positive integers $m,n$, it is known that $\alpha^{\frac 1n}+\beta^{\frac 1m}$ is a rational number. Prove that each of $\alpha^{\frac 1n}$ and $\beta^{\frac 1m}$ is a rational number.
Prove that for any $ p, q\in\mathbb{N}$ with $ q > 1$ the following inequality holds:
\[ \left\vert\pi\minus{}\frac{p}{q}\right\vert\ge q^{\minus{}42}.\]
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 $f(x)=\prod_{n=1}^{\infty} \left( 1 + \frac{x}{2^n} \right)$. Show that at the point $x=1$, $f(x)$ and all its derivatives are irrational.
Find all positive integers \(k\) for which there is an irrational \(\alpha>1\) and a positive integer \(N\) such that \(\left\lfloor\alpha^{n}\right\rfloor\) is a perfect square minus \(k\) for every integer \(n\) with \(n>N\).
Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.
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 $a, b$ and $c$ be positive real numbers such that $$a^2+ab+ac-bc = 0.$$
a) Show that if two of the numbers $a, b$ and $c$ are equal, then at least one of the numbers $a, b$ and $c$ is irrational.
b) Show that there exist infinitely many triples $(m, n, p)$ of positive integers such that $$m^2 + mn + mp -np = 0.$$
Do there exist real numbers $a$ and $b$ such that [list=a][*] $a+b$ is rational and $a^n +b^n $ is irrational for all $n \in \mathbb{N}$ with $n \ge 2$? [*] $a+b$ is irrational and $a^n +b^n $ is rational for all $n \in \mathbb{N}$ with $n \ge 2$?[/list]
Prove that for $\forall n>1$, $n\in \mathbb{N}$ , there exist infinitely many pairs of positive irrational numbers $a$ and $b$, such that $a^n=b$.
It is possible to show that $ \csc\frac{3\pi}{29}\minus{}\csc\frac{10\pi}{29}\equal{} 1.999989433...$. Prove that there are no integers $ j$, $ k$, $ n$ with odd $ n$ satisfying $ \csc\frac{j\pi}{n}\minus{}\csc\frac{k\pi}{n}\equal{} 2$.