Found problems: 56
Let $p$ be a prime number. For integers $r, s$ such that $rs(r^2 - s^2)$ is not divisible by $p$, let $f(r, s)$ denote the number of integers $n \in \{1, 2, \ldots, p - 1\}$ such that $\{rn/p\}$ and $\{sn/p\}$ are either both less than $1/2$ or both greater than $1/2$. Prove that there exists $N > 0$ such that for $p \geq N$ and all $r, s$, \[ \left\lceil \frac{p-1}{3} \right\rceil \le f(r, s) \le \left\lfloor \frac{2(p-1)}{3} \right\rfloor. \]
Let be two continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R} $ satisfying the following equations:
$$ \lim_{x\to\infty } f(x) =\infty =\lim_{x\to\infty } g(x) $$
Prove that there exists a divergent sequence $ \left( k_n \right)_{n\ge 1} $ of nonnegative integers which has the property that each term (function) of the sequence of functions $ \left( h_{n} \right)_{n\ge 1} :[0,\infty )\longrightarrow\mathbb{R} $ defined as
$$ h_{n} (x) =f\left( k_n+g(x) -\left\lfloor g(x) \right\rfloor \right) , $$
doesn't have limit at $ \infty . $
[i]Romeo Ilie[/i]
Solve in the set of real numbers the equation $ a^{[ x ]} +\log_a\{ x \} =x , $ where $ a $ is a real number from the interval $ (0,1). $
$ [] $ and $ \{\} $ [i]denote the floor, respectively, the fractional part.[/i]
Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .
Solve the equation $\{(x + 1)^3\} = x^3$, where $\{z\}$ is the fractional part of the number z, i.e. $\{z\} = z - [z]$.
Find the smallest real number $C$, such that for any positive integers $x \neq y$ holds the following:
$$\min(\{\sqrt{x^2 + 2y}\}, \{\sqrt{y^2 + 2x}\})<C$$
Here $\{x\}$ denotes the fractional part of $x$. For example, $\{3.14\} = 0.14$.
[i]Proposed by Anton Trygub[/i]
Let $n \geq 2$ be an integer. Positive reals $x_1, x_2, \cdots, x_n$ satisfy $x_1x_2 \cdots x_n = 1$.
Show: $$\{x_1\} + \{x_2\} + \cdots + \{x_n\} < \frac{2n-1}{2}$$
Where $\{x\}$ denotes the fractional part of $x$.
How many solutions satisfying the condition $1 < x < 5$ does the equation $\{x[x]\} = 0.5$ have?
(Here $[x]$ is the integer part of the number $x$, $\{x\} = x - [x]$ is the fractional part of the number $x$.)
Find all positive numbers $x$ such that:$$\frac{1}{[x]}-\frac{1}{[2x]}=\frac{1}{6\{x\}}$$ where $[x]$ represents the integer part of $x$ and $\{x\}=x-[x]$.
Prove that the application $ \mathbb{R}\ni x\mapsto 2x+ \{ x\} $ and its inverse are bijective and continuous.
Indicate (justifying your answer) if there exists a function $f: R \to R$ such that for all $x \in R$ fulfills that
i) $\{f(x))\} \sin^2 x + \{x\} cos (f(x)) cosx =f (x)$
ii) $f (f(x)) = f(x)$
where $\{m\}$ denotes the fractional part of $m$. That is, $\{2.657\} = 0.657$, and $\{-1.75\} = 0.25$.
Consider the sequence $(a_n)_{n\geqslant 1}$ defined by $a_1=1/2$ and $2n\cdot a_{n+1}=(n+1)a_n.$[list=a]
[*]Determine the general formula for $a_n.$
[*]Let $b_n=a_1+a_2+\cdots+a_n.$ Prove that $\{b_n\}-\{b_{n+1}\}\neq \{b_{n+1}\}-\{b_{n+2}\}.$
[/list]
For every positive integer $k>1$ prove that there exist a real number $x$ so that for every positive integer $n<1398$:
$$\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.$$
[i]Proposed by Mohammad Amin Sharifi[/i]
Find the real numbers $ x>1 $ having the property that $ \sqrt[n]{\lfloor x^n \rfloor } $ is an integer for any natural number $ n\ge 2. $
[i]Mihai Piticari[/i] and [i]Dan Popescu[/i]
Find all real $x > 0$ and integer $n > 0$ so that $$ \lfloor x \rfloor+\left\{ \frac{1}{x}\right\}= 1.005 \cdot n.$$
Prove that $\left\{ \frac{m}{n}\right\}+\left\{ \frac{n}{m}\right\} \ne 1$ , for any positive integers $m, n$.
Find all real numbers $a$ such that $3 < a < 4$ and $a(a-3\{a\})$ is an integer.
(Here $\{a\}$ denotes the fractional part of $a$.)
For any positive integer $n{}$ define $a_n=\{n/s(n)\}$ where $s(\cdot)$ denotes the sum of the digits and $\{\cdot\}$ denotes the fractional part.[list=a]
[*]Prove that there exist infinitely many positive integers $n$ such that $a_n=1/2.$
[*]Determine the smallest positive integer $n$ such that $a_n=1/6.$
[/list]
[i]Marius Burtea[/i]
Solve for real $x$ : $2^{2x} \cdot 2^{3\{x\}} = 11 \cdot 2^{5\{x\}} + 5 \cdot 2^{2[x]}$
(For a real number $x, [x]$ denotes the greatest integer less than or equal to x. For instance, $[2.5] = 2$, $[-3.1] = -4$, $[\pi ] = 3$. For a real number $x, \{x\}$ is defined as $x - [x]$.)
For natural number $n$ we define
\[
a_n = \{ \sqrt{n} \} - \{ \sqrt{n + 1} \} + \{ \sqrt{n + 2} \} - \{ \sqrt{n + 3} \}.
\]
a) Show that $a_1 > 0,2$.
b) Show that $a_n < 0$ for infinity many values of $n$ and $a_n > 0$ for infinity values of natural numbers of $n$ as well. ( We denote by $\{ x \} $ the fractional part of $x.$)
Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .
Positive integers $x, y$ satisfy the following conditions:
$$\{\sqrt{x^2 + 2y}\}> \frac{2}{3}; \hspace{10mm} \{\sqrt{y^2 + 2x}\}> \frac{2}{3}$$
Show that $x = y$.
Here $\{x\}$ denotes the fractional part of $x$. For example, $\{3.14\} = 0.14$.
[i]Proposed by Anton Trygub[/i]
Froa nay real $x$, we denote $[x]$, the integer part of $x$ and with $\{x\}$ the fractional part of $x$, such that $x=[x]+\{x\}$.
a) Find at least one real $x$ such that$\{x\}+\left\{\frac{1}{x}\right\}=1$
b) Find all rationals $x$ such that $\{x\}+\left\{\frac{1}{x}\right\}=1$
Solve the equation $[x]\{x\} = 1999x$, where $[x]$ denotes the largest integer less than or equal to $x$, and $\{x\} = x -[x] $
For every positive integer $k>1$ prove that there exist a real number $x$ so that for every positive integer $n<1398$:
$$\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.$$
[i]Proposed by Mohammad Amin Sharifi[/i]