Found problems: 102
A positive rational number is written on the blackboard. Every minute Vasya replaces the number $ r $ written on the board with $ \sqrt {r + 1} $. Prove that someday he will get an irrational number.
Let $\sqrt 3 = 1.b_1b_2b_3 \dots _{(2)}$ be the binary representation of $\sqrt 3$. Prove that for any positive integer $n$, at least one of the digits $b_n$, $b_{n+1}$, $\dots$, $b_{2n}$ equals $1$.
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.
Let $a_0$ be a positive integer and $a_{n + 1} =\sqrt{a_n^2 + 1}$, for all $n \ge 0$.
1) Prove that for all $a_0$ the sequence contains infinitely many integers and infinitely many irrational numbers.
2) Is there an $a_0$ for which $a_{2010}$ is an integer?
Show that the number
\[\sqrt[n]{\sqrt{2019} + \sqrt{2018}} + \sqrt[n]{\sqrt{2019} - \sqrt{2018}}\]
is irrational for any $n\ge 2$.
Prove that for any irrational number $\xi$, there are infinitely many rational numbers $\frac{m}{n}$ $\left( (m,n) \in \mathbb{Z}\times \mathbb{N}\right)$ such that \[\left\vert \xi-\frac{n}{m}\right\vert < \frac{1}{\sqrt{5}m^{2}}.\]
Suppose that $p, q \in \mathbb{N}$ satisfy the inequality \[\exp(1)\cdot( \sqrt{p+q}-\sqrt{q})^{2}<1.\] Show that $\ln \left(1+\frac{p}{q}\right)$ is irrational.
Let $ a,b$ be real nonzero numbers, such that number $ \lfloor an \plus{} b \rfloor$ is an even integer for every $ n \in \mathbb{N}$. Prove that $ a$ is an even integer.
Let $p(x)=x^{3}+a_{1}x^{2}+a_{2}x+a_{3}$ have rational coefficients and have roots $r_{1}$, $r_{2}$, and $r_{3}$. If $r_{1}-r_{2}$ is rational, must $r_{1}$, $r_{2}$, and $r_{3}$ be rational?
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.
Prove that for any real numbers $ a_1, a_2, \dots, a_ {100} $ there exists a real number $ b $ such that all numbers $ a_i + b $ ($ 1 \leq i \leq 100 $) are irrational.
$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such
\[ \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}.\]
Prove that $ c_n \equal{} 0$ implies $ n \equal{} 0.$
Let $\{a_{n}\}_{n \ge 1}$ be a sequence of positive numbers such that \[a_{n+1}^{2}= a_{n}+1, \;\; n \in \mathbb{N}.\] Show that the sequence contains an irrational number.
Is $\log_{10}{8}$ rational?
Let $r$ be a natural number greater than $1$. Then there exist positive irrational numbers $x, y$ such that $x^y = r$ . Prove it.
Let $\alpha$ be an irrational number with $0<\alpha < 1$, and draw a circle in the plane whose circumference has length $1$. Given any integer $n\ge 3$, define a sequence of points $P_1, P_2, \ldots , P_n$ as follows. First select any point $P_1$ on the circle, and for $2\le k\le n$ define $P_k$ as the point on the circle for which the length of arc $P_{k-1}P_k$ is $\alpha$, when travelling counterclockwise around the circle from $P_{k-1}$ to $P_k$. Suppose that $P_a$ and $P_b$ are the nearest adjacent points on either side of $P_n$. Prove that $a+b\le n$.
Find all irrational real numbers $\alpha$ such that
\[ \alpha^3 - 15 \alpha \text{ and } \alpha^4 - 56 \alpha \]
are both rational numbers.
a) Show that there exist irrational numbers $a$, $b$, and $c$ such that the numbers $a+b\cdot c$, $b+a\cdot c$, and $c+a\cdot b$ are rational numbers.
b) Show that if $a$, $b$, and $c$ are real numbers such that $a+b+c=1$, and the numbers $a+b\cdot c$, $b+a\cdot c$, and $c+a\cdot b$ are rational and non-zero, then $a$, $b$, and $c$ are rational numbers.
There are $6$ irrational numbers. Prove that there are always three of them, suppose $a,b,c$ such that $a+b$,$b+c$,$c+a$ are irrational numbers.
[i](Erken)[/i]
It is known that for irrational numbers $\alpha$, $\beta$, $\gamma$, $\delta$ and for any positive integer $n$ the following is true:
$$[n\alpha]+[n\beta]=[n\gamma]+[n\delta]$$
Does this mean that sets $\{\alpha,\beta\}$ and $\{\gamma,\delta\}$ are equal? (As usual $[x]$ means the greatest integer not greater than $x$).
Given six irrational numbers, will it be possible to choose three such that the sum of any two of these three 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:
Find all integers $n\ge 3$ for which the following statement is true:
Any arithmetic progression $a_1,\ldots ,a_n$ with $n$ terms for which $a_1+2a_2+\ldots+na_n$ is rational contains at least one rational term.
Let the irrational number
\[\alpha =1-\cfrac{1}{2a_1-\cfrac{1}{2a_2-\cfrac{1}{2a_3-\cdots}}}\]
where coefficients $a_1, a_2, \ldots$ are positive integers, infinitely many of which are greater than $1$. Prove that for every positive integer $N$ at least half of the numbers $\lfloor \alpha\rfloor, \lfloor 2\alpha\rfloor, \ldots, \lfloor N\alpha\rfloor$ are even.
[i]Proposed by Géza Kós, Budapest[/i]
In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are rational numbers, we call it rational point. If $a$ is an irrational number, then in all lines that passes $(a,0)$,
$\text{(A)}$There are infinitely many lines, on which there are at least two rational points.
$\text{(B)}$There are exactly $n(n\geq2)$ lines, on which there are at least two rational points.
$\text{(C)}$There are exactly 1 line, on which there are at least two rational points.
$\text{(D)}$Every line passes at least one rational point.