Found problems: 4776
Find the functions $f:\mathbb{Z}\times \mathbb{Z}\to\mathbb{R}$ such that
a) $f(x,y)\cdot f(y,z) \cdot f(z,x) = 1$ for all integers $x,y,z$;
b) $f(x+1,x)=2$ for all integers $x$.
Find all functions $f: \mathbb{Q}\to \mathbb{Q}$ such that for all $x,y,z \in \mathbb{Q}$: \[f(x+y+z)+f(x-y)+f(y-z)+f(z-x)=3f(x)+3f(y)+3f(z).\]
Consider a function $f:\mathbb{Z}\to \mathbb{Z}$ such that:
\[f(m^2+f(n))=f^2(m)+n,\ \forall m,n\in \mathbb{Z}\]
Prove that:
a)$f(0)=0$;
b)$f(1)=1$;
c)$f(n)=n,\ \forall n\in \mathbb{Z}$
[i]Lucian Dragomir[/i]
Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which
\[ x(f(x+1)-f(x)) = f(x), \]
for all $x\in\mathbb{R}$ and
\[ | f(x) - f(y) | \leq |x-y| , \]
for all $x,y\in\mathbb{R}$.
[i]Mihai Piticari[/i]
For a positive constant $ t$, let $ \alpha ,\ \beta$ be the roots of the quadratic equation $ x^2 \plus{} t^2x \minus{} 2t \equal{} 0$.
Find the minimum value of $ \int_{ \minus{} 1}^2 \left\{\left(x \plus{} \frac {1}{\alpha ^ 2}\right)\left(x \plus{} \frac {1}{\beta ^ 2}\right) \plus{} \frac {1}{\alpha \beta}\right\}dx.$
The semicircle with centre $O$ and the diameter $AC$ is divided in two arcs $AB$ and $BC$ with ratio $1: 3$. $M$ is the midpoint of the radium $OC$. Let $T$ be the point of arc $BC$ such that the area of the cuadrylateral $OBTM$ is maximum. Find such area in fuction of the radium.
Let be a real number $ a\in (0,1) $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with the property that:
$$ \lim_{x\to 0} f(x) =0= \lim_{x\to 0} \frac{f(x)-f(ax)}{x} $$
Prove that $ \lim_{x\to\infty } \frac{f(x)}{x} =0. $
Let $f$ be a bijective function from $A = \{ 1, 2, \ldots, n \}$ to itself. Show that there is a positive integer $M$ such that $f^{M}(i) = f(i)$ for each $i$ in $A$, where $f^{M}$ denotes the composition $f \circ f \circ \cdots \circ f$ $M$ times.
Define a function $f(x)=\int_0^{\frac{\pi}{2}} \frac{\cos |t-x|}{1+\sin |t-x|}dt$ for $0\leq x\leq \pi$.
Find the maximum and minimum value of $f(x)$ in $0\leq x\leq \pi$.
The three roots of $P(x) = x^3 - 2x^2 - x + 1$ are $a>b>c \in \mathbb{R}$. Find the value of $a^2b+b^2c+c^2a$. :D
Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and
\[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\]
Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$
Prove that $ f(x)\equal{}\int_0^1 e^{\minus{}|t\minus{}x|}t(1\minus{}t)dt$ has maximal value at $ x\equal{}\frac 12$.
Determine all integers $k$ for which there exists a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}$ such that $f(2023) = 2024$ and $f(ab) = f(a) + f(b) + kf(\gcd(a,b))$ for all positive integers $a$ and $b$.
Find the mean value of the function $\ln r$ on the circle $(x - a)^2 + (y-b)^2 = R^2$ (of the function $1/r$ on the sphere).
Let $n$ be a nonzero natural number and $f:\mathbb R\to\mathbb R\setminus\{0\}$ be a function such that $f(2014)=1-f(2013)$. Let $x_1,x_2,x_3,\ldots,x_n$ be real numbers not equal to each other. If
$$\begin{vmatrix}1+f(x_1)&f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&1+f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&f(x_2)&1+f(x_3)&\cdots&f(x_n)\\\vdots&\vdots&\vdots&\ddots&\vdots\\f(x_1)&f(x_2)&f(x_3)&\cdots&1+f(x_n)\end{vmatrix}=0,$$prove that $f$ is not continuous.
Find all the functions $f: \mathbb{Z}\to \mathbb{Z}$ satisfying the following property: if $a$, $b$ and $c$ are integers such that $a+b+c=0$, then
$$f(a)+f(b)+f(c)=a^2+b^2+c^2.$$
Find the total number of different integer values the function \[f(x) = \lfloor x\rfloor+\lfloor 2x\rfloor+\left\lfloor \frac{5x}{3}\right\rfloor+\lfloor 3x\rfloor+\lfloor 4x\rfloor\] takes for real numbers $x$ with $0 \leq x \leq 100$.
Let $m$ be a positive integer and $\{a_n\}_{n\geq 0}$ be a sequence given by $a_0 = a \in \mathbb N$, and \[ a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + m & \textrm{ otherwise. } \end{cases} \]
Find all values of $a$ such that the sequence is periodical (starting from the beginning).
The points that satisfy the system $ x \plus{} y \equal{} 1,\, x^2 \plus{} y^2 < 25,$ constitute the following set:
$ \textbf{(A)}\ \text{only two points} \qquad \\
\textbf{(B)}\ \text{an arc of a circle}\qquad \\
\textbf{(C)}\ \text{a straight line segment not including the end\minus{}points}\qquad \\
\textbf{(D)}\ \text{a straight line segment including the end\minus{}points}\qquad \\
\textbf{(E)}\ \text{a single point}$
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following relationship for all real numbers $x$ and $y$\[f(xf(y)-f(x))=2f(x)+xy.\]
For every $x \ge -\frac{1}{e}\,$, there is a unique number $W(x) \ge -1$ such that
\[
W(x) e^{W(x)} = x.
\]
The function $W$ is called Lambert's $W$ function. Let $y$ be the unique positive number such that
\[
\frac{y}{\log_{2} y} = - \frac{3}{5} \, .
\]
The value of $y$ is of the form $e^{-W(z \ln 2)}$ for some rational number $z$. What is the value of $z$?
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ for which the equality
\[f(f(x)\minus{}y)\equal{}f(x)\plus{}f(f(y)\minus{}f(\minus{}x))\plus{}x\]
holds for all real $x,y$.
Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.
For $ x \ge 0$ the smallest value of $ \frac {4x^2 \plus{} 8x \plus{} 13}{6(1 \plus{} x)}$ is:
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \frac {25}{12} \qquad \textbf{(D)}\ \frac {13}{6} \qquad \textbf{(E)}\ \frac {34}{5}$
Consider real numbers $x,y,z$ such that $x,y,z>0$. Prove that \[ (xy+yz+xz)\left(\frac{1}{x^2+y^2}+\frac{1}{x^2+z^2}+\frac{1}{y^2+z^2}\right) > \frac{5}{2}. \]