Found problems: 56
Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$, one has
$$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$$
where $\{ x \}$ denotes the fractional part of $x$.
For this exercise, $ \{\} $ denotes the fractional part.
[b]a)[/b] Let be a natural number $ n. $ Compare $ \left\{ \sqrt{n+1} -\sqrt{n} \right\} $ with $ \left\{ \sqrt{n} -\sqrt{n-1} \right\} . $
[b]b)[/b] Show that there are two distinct natural numbers $ a,b, $ such that $ \left\{ \sqrt{a} -\sqrt{b} \right\} =\left\{ \sqrt{b} -\sqrt{a} \right\} . $
[i]Traian Preda[/i]
A positive rational number $x$ is called [i]banzai [/i] if the following conditions are met:
$\bullet$ $x=\frac{p}{q}>1$ where $p,q$ are comprime natural numbers
$\bullet$ exist constants $\alpha,N$ such that for all integers $n\geq N$,$$\mid \left\{\,x^n\right\} -\alpha\mid \leq \dfrac{1}{2(p+q)}.$$
Find the total number of banzai numbers.
Note:$\left\{\,x\right\}$ means fractional part of $x$
[i](tatari/nightmare)[/i]
Let $ a, b, c$ be positive integers satisfying the conditions $ b > 2a$ and $ c > 2b.$ Show that there exists a real number $ \lambda$ with the property that all the three numbers $ \lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $ \left(\frac {1}{3}, \frac {2}{3} \right].$
A positive integer $x$ satisfies the following:
\[\{\frac{x}{3}\}+\{\frac{x}{5}\}+\{\frac{x}{7}\}+\{\frac{x}{11}\}=\frac{248}{165}\]
Find all possible values of
\[\{\frac{2x}{3}\}+\{\frac{2x}{5}\}+\{\frac{2x}{7}\}+\{\frac{2x}{11}\}\]
where $\{y\}$ denotes the fractional part of $y$.
Let $p > 2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$, such that
\[ \left\{ \dfrac{ra}{p} \right\} + \left\{ \dfrac{rb}{p} \right\} + \left\{ \dfrac{rc}{p} \right\} + \left\{ \dfrac{rd}{p} \right\} = 2 \]
for any integer $r$ not divisible by $p$. Prove that at least two of the numbers $a+b$, $a+c$, $a+d$, $b+c$, $b+d$, $c+d$ are divisible by $p$.
(Note: $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.)
Let ${a}$ be the fractional part of the number $a$, that is, $\{a\} = a - [a]$, where$ [a]$ is the integer part of $ a$. (For example, $\{1.7\} = 1.7 -1 = 0.7$,$\{-\sqrt2 \}= -\sqrt2 -(-3) = 3-\sqrt2$.)
a) How many solutions does the equation have
$$ \{5\{4\{3\{2\{x\}\}\}\}\}=1\,\, ?$$
b) Find its greatest solution.
If $ a,n $ are two natural numbers corelated by the equation $ \left\{ \sqrt{n+\sqrt n}\right\} =\left\{ \sqrt a\right\} , $ then $
1+4a $ is a perfect square. Justify this statement. Here, $ \{\} $ is the usual fractionary part.
Solve the system of equations:
$$\begin{cases} x + [y] + \{z\}=3.9 \\
y + [z] + \{x\}= 3.5 \\
z + [x] + \{y\}= 2.
\end{cases}$$
The integer part $I (A)$ of a number $A$ is the greatest integer which is not greater than $A$ , while the fractional part $F(A)$ is defined as $A - I(A)$ .
(a) Give an example of a positive number $A$ such that $F(A) + F( 1/A) = 1$ .
(b) Can such an $A$ be a rational number?
(I. Varge, Romania)
Solve in the set of real numbers the equation
$$ 16\{ x \}^2-8x=-1, $$
where $ \{\} $ denotes the fractional part.
Let $a$ and $b$ be two numbers in the interval $(0,1)$ such that $a$ is rational and
[center]$\{na\} \ge \{nb\},$ for every nonnegative integer $n.$[/center]
Prove that $a=b.$
(Note: $\{x\}$ is the fractional part of $x.$)
[b]a)[/b] Prove that $ \{ a \} +\{ 1/a \} <3/2, $ for any positive real number $ a. $
[b]b)[/b] Give an example of a number $ b $ satisfying $ \{ b \} +\{ 1/b \} =1. $
[i]{} means fractional part[/i]
Let $n\geq 3$ be a natural number and $x\in \mathbb{R}$, for which $\{ x\} =\{ x^2\} =\{ x^n\} $ (with $\{ x\} $ we denote the fractional part of $x$). Prove that $x$ is an integer.
Find the smallest positive root of the equation $$\{tg x\}=\sin x. $$
($\{a\}$ is the fractional part of $a$, $\{a\}$ is equal to the difference between $ a$ and the largest integer not exceeding $a$.)
Let be a nonzero real number $ a, $ and a natural number $ n. $ Prove the implication:
$$ \{ a \} +\left\{\frac{1}{a}\right\} =1 \implies \{ a^n \} +\left\{\frac{1}{a^n}\right\} =1 , $$
where $ \{\} $ is the fractional part.
[b]a)[/b] Show that there are infinitely many rational numbers $ x>0 $ such that $ \left\{ x^2 \right\} +\{ x \} =0.99. $
[b]b)[/b] Show that there are no rational numbers $ x>0 $ such that $ \left\{ x^2 \right\} +\{ x \} =1. $
$ \{\} $ denotes the usual fractional part.
For a real number $ k\ge 2, $ solve the equation $ \frac{\{x\}[x]}{x} =k. $
Let $n$ be a positive integer. Prove that $n\sqrt{19}\{n\sqrt{19}\} > 1$, where $\{x\}$ denotes the fractional part of $x$.
Solve in $ \mathbb{R} $ the equation $ [x]^5+\{ x\}^5 =x^5, $ where $ [],\{\} $ are the integer part, respectively, the fractional part.
Given a real number $x$, define the series
\[
S(x) = \sum_{n=1}^{\infty} \{n! \cdot x\},
\]
where $\{s\} = s - \lfloor s \rfloor$ is the fractional part of the number $s$. Determine if there exists an irrational number $x$ for which the series $S(x)$ converges.
Prove that for every positive integer $n$ holds inequality $\{n\sqrt{7}\}>\frac{3\sqrt{7}}{14n}$, where $\{x\}$ is fractional part of $x$.
Let $n\geq 3$ be a natural number and $x\in \mathbb{R}$, for which $\{ x\} =\{ x^2\} =\{ x^n\} $ (with $\{ x\} $ we denote the fractional part of $x$). Prove that $x$ is an integer.
Solve in the real numbers the equation $ 2^{1+x} =2^{[x]} +2^{\{x\}} , $ where $ [],\{\} $ deonotes the ineger and fractional part, respectively.
[i]Aurel Bârsan[/i]
Show that there is a natural number $ n $ that satisfies the following inequalities:
$$ \sqrt{3} -\frac{1}{10}<\{ n\sqrt 3\} +\{ (n+1)\sqrt 3 \} <\sqrt 3. $$