Let $\alpha=0.d_{1}d_{2}d_{3} \cdots$ be a decimal representation of a real number between $0$ and $1$. Let $r$ be a real number with $\vert r \vert<1$. [list=a][*] If $\alpha$ and $r$ are rational, must $\sum_{i=1}^{\infty} d_{i}r^{i}$ be rational? [*] If $\sum_{i=1}^{\infty} d_{i}r^{i}$ and $r$ are rational, $\alpha$ must be rational? [/list]

If $ x \plus{} y \equal{} 1$, then the largest value of $ xy$ is: $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 0.5\qquad \textbf{(C)}\ \text{an irrational number about }{0.4}\qquad \textbf{(D)}\ 0.25\qquad \textbf{(E)}\ 0$

For a non-empty set $T$ denote by $p(T)$ the product of all elements of $T$. Does there exist a set $T$ of $2021$ elements such that for any $a\in T$ one has that $P(T)-a$ is an odd integer? Consider two cases: 1) All elements of $T$ are irrational numbers. 2) At least one element of $T$ is a rational number.

For which angles $ \theta$, with $ \theta$ a rational number of degrees, is $ {\tan}^{2}\theta\plus{}{\tan}^{2}2\theta$ is irrational?

Let $A$ be an infinite set of real numbers containing at least one irrational number. Prove that for every natural number $n > 1$ there exists a subset $S$ of $A$ with n elements such that the sum of the elements of $S$ is an irrational number.

The set $ S$ is a finite subset of $ [0,1]$ with the following property: for all $ s\in S$, there exist $ a,b\in S\cup\{0,1\}$ with $ a, b\neq s$ such that $ s \equal{}\frac{a\plus{}b}{2}$. Prove that all the numbers in $ S$ are rational.

Show that $\frac{1}{\pi} \arccos \left( \frac{1}{\sqrt{2003}} \right)$ 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.

Given six irrational numbers, will it be possible to choose three such that the sum of any two of these three is irrational?

Show that the number \[\sqrt[n]{\sqrt{2019} + \sqrt{2018}} + \sqrt[n]{\sqrt{2019} - \sqrt{2018}}\] is irrational for any $n\ge 2$.

Real numbers $x_1, x_2, x_3, x_4$ are roots of the fourth degree polynomial $W (x)$ with integer coefficients. Prove that if $x_3 + x_4$ is a rational number and $x_3x_4$ is a irrational number, then $x_1 + x_2 = x_3 + x_4$.

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.

Let $(a_{n})_{n \ge 1}$ be a sequence of integers satisfying the inequality \[ 0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1 \] for all $n \ge 2$. Prove that the sequence $(a_{n})$ is periodic. Any Hints or Sols for this hard problem?? :help:

[b]a)[/b] Give example of two irrational numbers $ a,b $ having the property that $ a^3,b^3,a+b $ are all rational. [b]b)[/b] Prove that if $ x,y $ are two nonnegative real numbers having the property that $ x^3,y^3,x+y $ are rational, then $ x $ and $ y $ are both rational. [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

Is $\log_{10}{8}$ rational?

Show that $\cos 1^{\circ}$ is irrational.

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$.

Show that $\tan \left( \frac{\pi}{m} \right)$ is irrational for all positive integers $m \ge 5$.

Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$, \[[r^m] \equiv -1 \pmod k .\] [i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients. [i]Proposed by Yugoslavia.[/i]

Let $f:\mathbb{Q}\rightarrow \mathbb{Q}$ a monotonic bijective function. a)Prove that there exist a unique continuous function $F:\mathbb{R}\rightarrow \mathbb{R}$ such that $F(x)=f(x),\ (\forall)x\in \mathbb{Q}$. b)Give an example of a non-injective polynomial function $G:\mathbb{R}\rightarrow \mathbb{R}$ such that $G(\mathbb{Q})\subset \mathbb{Q}$ and it's restriction defined on $\mathbb{Q}$ is injective.

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).

[b]1.[/b] For each real number $r$ between $0$ and $1$ we can represent $r$ as an infinite decimal $r = 0.r_1r_2r_3\dots$ with $0 \leq r_i \leq 9$. For example, $\frac{1}{4} = 0.25000\dots$, $\frac{1}{3} = 0.333\dots$ and $\frac{1}{\sqrt{2}} = 0.707106\dots$. a) Show that we can choose two rational numbers $p$ and $q$ between $0$ and $1$ such that, from their decimal representations $p = 0.p_1p_2p_3\dots$ and $q = 0.q_1q_2q_3\dots$, it's possible to construct an irrational number $\alpha = 0.a_1a_2a_3\dots$ such that, for each $i = 1, 2, 3, \dots$, we have $a_i = p_1$ or $a_1 = q_i$. b) Show that there's a rational number $s = 0.s_1s_2s_3\dots$ and an irrational number $\beta = 0.b_1b_2b_3\dots$ such that, for all $N \geq 2017$, the number of indexes $1 \leq i \leq N$ satisfying $s_i \neq b_i$ is less than or equal to $\frac{N}{2017}$.

In each cell of a chessboard with $2$ rows and $2019$ columns a real number is written so that: [LIST] [*] There are no two numbers written in the first row that are equal to each other.[/*] [*] The numbers written in the second row coincide with (in some another order) the numbers written in the first row.[/*] [*] The two numbers written in each column are different and they add up to a rational number.[/*] [/LIST] Determine the maximum quantity of irrational numbers that can be in the chessboard.

An integer-sided triangle has angles $ p\theta$ and $ q\theta$, where $ p$ and $ q$ are relatively prime integers. Prove that $ \cos\theta$ is irrational.

You are given three lists A, B, and C. List A contains the numbers of the form $10^{k}$ in base 10, with $k$ any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively: \[\begin{array}{lll}A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}.\] Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists B or C that has exactly $n$ digits.