Found problems: 27
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]
Determine the real numbers $ a,b, $ such that
$$ [ax+by]+[bx+ay]=(a+b)\cdot [x+y],\quad\forall x,y\in\mathbb{R} , $$
where $ [t] $ is the greatest integer smaller than $ t. $
For arbitrary natural number $n$, prove that $\sum^n_{k=0}C^k_n2^kC^{[(n-k)/2]}_{n-k}=C^n_{2n+1}$, where $C^0_0=1$ and $[\dfrac{n-k}{2}]$ denotes the integer part of $\dfrac{n-k}{2}$.
Suppose $x$ is a positive real number such that $\{x\}, [x]$ and $x$ are in a geometric progression. Find the least positive integer $n$ such that $x^n > 100$. (Here $[x]$ denotes the integer part of $x$ and $\{x\} = x - [x]$.)
Let $a, b$ and $c$ be positive integers such that $a^2 + b^2 + 1 = c^2$ . Prove that $[a/2] + [c / 2]$ is even.
Note: $[x]$ is the integer part of $x$.
Solve the equation in the set of real numbers:
$$\left[ x+\frac{1}{x} \right] = \left[ x^2+\frac{1}{x^2} \right]$$
where $[a]$, represents the integer part of the real number $a$.
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function with the property that $ (f\circ f) (x) =[x], $ for any real number $ x. $ Show that there exist two distinct real numbers $ a,b $ so that $ |f(a)-f(b)|\ge |a-b|. $
$ [] $ denotes the integer part.
The numerical sequence $x_1 , x_2 ,.. $ satisfies $x_1 = \frac12$ and $x_{k+1} =x^2_k+x_k$ for all natural integers $k$ . Find the integer part of the sum $\frac{1}{x_1+1}+\frac{1}{x_2+1}+...+\frac{1}{x_{100}+1}$
{A. Andjans, Riga)
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 numbers $k $ such that $x^2 - 2 x [x] + x - k = 0$ has at least two non-negative roots.
Find all real $x > 0$ and integer $n > 0$ so that $$ \lfloor x \rfloor+\left\{ \frac{1}{x}\right\}= 1.005 \cdot n.$$
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 $x - [x]$.)
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 $\{x\}+\left\{\frac{1}{x}\right\}=1$
Find all positive integers $m$ such that
$$\{\sqrt{m}\} = \{\sqrt{m+ 2011}\}.$$
The $n$-th digit of number $a = 0.12457...$ equals the first digit of the integer part of the number $n\sqrt2$. Prove that $a$ is irrational number. (6)
For a natural number $ n\ge 2, $ calculate the integer part of $ \sqrt[n]{1+n}-\sqrt {2/n} . $
[i]Dan Nedeianu[/i]
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)
For a real number $ k\ge 2, $ solve the equation $ \frac{\{x\}[x]}{x} =k. $
Solve in $ \mathbb{R} $ the equation $ [x]^5+\{ x\}^5 =x^5, $ where $ [],\{\} $ are the integer part, respectively, the fractional part.
Find all the real numbers $x$ such that $\frac{1}{[x]}+\frac{1}{[2x]}=\{x\}+\frac{1}{3}$ where $[x]$ denotes the integer part of $x$ and $\{x\}=x-[x]$.
For example, $[2.5]=2, \{2.5\} = 0.5$ and $[-1.7]= -2, \{-1.7\} = 0.3$
Consider the parametric function $ f_k:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=x+k\lfloor x \rfloor . $
[b]a)[/b] For which integer values of $ k $ the above function is injective?
[b]b)[/b] For which integer values of $ k $ the above function is surjective?
[b]c)[/b] Given two natural numbers $ n,m, $ create two bijective functions:
$$ \phi : f_m (\mathbb{R} )\cap [0,\infty )\longrightarrow f_n(\mathbb{R})\cap [0,\infty ) $$
$$ \psi : \left(\mathbb{R}\setminus f_m (\mathbb{R})\right)\cap [0,\infty )\longrightarrow\left(\mathbb{R}\setminus f_n (\mathbb{R})\right)\cap [0,\infty ) $$
[i]Cristinel Mortici[/i]
Given an integer $n\ge 2$, prove that
\[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\].
[hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)[/hide]
Let $r$ be a positive real number. Denote by $[r]$ the integer part of $r$ and by $\{r\}$ the fractional part of $r$. For example, if $r=32.86$, then $\{r\}=0.86$ and $[r]=32$. What is the sum of all positive numbers $r$ satisfying $25\{r\}+[r]=125$?
Find all the real numbers $x$ such that $\frac{1}{[x]}+\frac{1}{[2x]}=\{x\}+\frac{1}{3}$ where $[x]$ denotes the integer part of $x$ and $\{x\}=x-[x]$.
For example, $[2.5]=2, \{2.5\} = 0.5$ and $[-1.7]= -2, \{-1.7\} = 0.3$
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]