Found problems: 2349
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $$\displaystyle{f(f(x)) = \frac{x^2 - x}{2}\cdot f(x) + 2-x,}$$ for all $x \in \mathbb{R}.$ Find all possible values of $f(2).$
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
$$f(xf(y)-y^2)=(y+1)f(x-y)$$
holds for all $x,y\in\mathbb{R}$.
Find all ordered pairs of integers $(a,b)$ such that there exists a function $f\colon \mathbb{N} \to \mathbb{N}$ satisfying
$$f^{f(n)}(n)=an+b$$
For all $n\in \mathbb{N}$.
Does there exist a function $f : \mathbb{R} \rightarrow \mathbb{R}$, which satisfies both conditions :
[b]a)[/b] $f( x + y + z) \leq 3(xy + yz + zx)$ for all real numbers $x , y , z$
and
[b]b)[/b] there exist function $g$ and natural number $n$, such that
$g(g(x)) = x ^ {2n + 1}$ and $f(g(x)) = (g(x)) ^2$ for every real number $x$ ?
Let $T$ the set of the infinite sequences of integers. For two given elements in $T$:
$(a_{1},a_{2},a_{3},...)$ and $(b_{1},b_{2},b_{3},...)$, define the sum
$(a_{1},a_{2},a_{3},...)+(b_{1},b_{2},b_{3},...)=(a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3},...)$.
Let $f: T\rightarrow$ $\mathbb{Z}$ a function such that:
i) If $x\in T$ has exactly one of your terms equal $1$ and all the others equal $0$, then $f(x)=0$.
ii)$f(x+y)=f(x)+f(y)$, for all $x,y\in T$.
Prove that $f(x)=0$ for all $x\in T$
Given a number $n\in\mathbb{Z}^+$ and let $S$ denotes the set $\{0,1,2,...,2n+1\}$. Consider the function $f:\mathbb{Z}\times S\to [0,1]$ satisfying two following conditions simultaneously:
i) $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$;
ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x\in\mathbb{Z}$ and $y\in\{1,2,3,...,2n\}$.
Let $F$ be the set of such functions. For each $f\in F$, let $v(f)$ be the set of values of $f$.
a) Proof that $|F|=\infty$.
b) Proof that for each $f\in F$ then $|v(f)|<\infty$.
c) Find the maximum value of $|v(f)|$ for $f\in F$.
Find a continuous function $f(x)$ satisfying the identity $f(x)-f(ax)=x^n-x^m$, where $n,m\in N , 0<a<1$
Prove that the functional equations
\[f(x + y) = f(x) + f(y),\]
\[ \text{and} \qquad f(x + y + xy) = f(x) + f(y) + f(xy) \quad (x, y \in \mathbb R)\]
are equivalent.
A function $f$ satisfies $f(x)+x f(1-x) = x^2$ for all real $x$. Determine $f$ .
A function $ f$ defined on the positive integers (and taking positive integers values) is given by:
$ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\
f(2 \cdot n) \equal{} f(n) \\
f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\
f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$
for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Find all functions $f :\mathbb{N} \rightarrow \mathbb{N}$ such that
$f(m + f(n)f(m)) = nf(m) + m$
holds for all $m,n \in \mathbb{N}$.
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that for all integers $a$ and $b$, we have: $$f(a^2+ab)+f(b^2+ab)=(a+b)f(a+b).$$
Proposed by Heidar Shushtari
Let \( \mathbb R \) be the set of real numbers. Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that, for all real numbers \( x \) and \( y \), the following equation holds:$$\big (x^2-y^2\big )f\big (xy\big )=xf\big (x^2y\big )-yf\big (xy^2\big ).$$
Let $\mathbb{N} = \{1, 2, 3, \dots\}$ and let $f : \mathbb{N}\to\mathbb{R}$. Prove that there is an infinite subset $A$ of $\mathbb{N}$ such that $f$ is increasing on $A$ or $f$ is decreasing on $A$.
Find all functions $f$ from the real numbers to the real numbers such that $f(xy) \le \frac12 \left(f(x) + f(y) \right)$ for all real numbers $x$ and $y$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)+f(x+y)=xy$ for all real numbers $x$ and $y$.
Let $ S$ be the set of all real numbers strictly greater than −1. Find all functions $ f: S \to S$ satisfying the two conditions:
(a) $ f(x \plus{} f(y) \plus{} xf(y)) \equal{} y \plus{} f(x) \plus{} yf(x)$ for all $ x, y$ in $ S$;
(b) $ \frac {f(x)}{x}$ is strictly increasing on each of the two intervals $ \minus{} 1 < x < 0$ and $ 0 < x$.
Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, such that
1) $f(0,x)$ is non-decreasing ;
2) for any $x,y \in \mathbb{R}$, $f(x,y)=f(y,x)$ ;
3) for any $x,y,z \in \mathbb{R}$, $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ;
4) for any $x,y,a \in \mathbb{R}$, $f(x+a,y+a)=f(x,y)+a$ .
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$.
[i]George Xing.[/i]
Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ such that for all integers $x$, $y$, $$f(x-f(y))=f(f(y))+f(x-2y)$$
[i]Proposed by Ivan Chan Kai Chin[/i]
The real function $f$ is defined for $\forall$ $x\in \mathbb{R}$ and $f(0)=0$. Also $f(9+x)=f(9-x)$ and $f(x-10)=f(-x-10)$ for $\forall$ $x\in \mathbb{R}$. What’s the least number of zeros $f$ can have in the interval $[0;2014]$? Does this change, if $f$ is also continuous?
Find all functions $f: R\to R$, which satisfy the equation $f (xy + f (x)) = xf (y) + f (x)$.
Find all functions $f : Z \to Z$ satisfying the following two conditions:
(i) for all integers $x$ we have $f(f(x)) = x$,
(ii) for all integers $x$ and y such that $x + y$ is odd, we have $f(x) + f(y) \ge x + y$.