Found problems: 1513
Let $ A $ be a subset of $ \mathbb{R} $ and let be a function $ f:A\longrightarrow\mathbb{R} $ satisfying
$$ f(x)-f(y)=(y-x)f(x)f(y),\quad\forall x,y\in A. $$
[b]a)[/b] Show that if $ A=\mathbb{R}, $ then $ f=0. $
[b]b)[/b] Find $ f, $ provided that $ A=\mathbb{R}\setminus\{1\} . $
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 $n\in \mathbb{N}$.
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.$
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 $f:\mathbb{R}\to\mathbb{R}$ 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 $f: R\to R$, which satisfy the equation $f (xy + f (x)) = xf (y) + f (x)$.
Let $ f(x)$ be a monic polynomial of degree $ 1991$ with integer coefficients. Define $ g(x) \equal{} f^2(x) \minus{} 9.$ Show that the number of distinct integer solutions of $ g(x) \equal{} 0$ cannot exceed $ 1995.$
Find all functions $f:\mathbb {Z}\to\mathbb Z$, satisfy that for any integer ${a}$, ${b}$, ${c}$,
$$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$
Let $\mathbb{Q}^+$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^+\to \mathbb{Q}^+$ satisfying$$f(x^2f(y)^2)=f(x^2)f(y)$$for all $x,y\in\mathbb{Q}^+$
Denote by $\mathbb{S}$ the set of all proper subsets of $\mathbb{Z}_{>0}$. Find all functions $f : \mathbb{S} \mapsto \mathbb{Z}_{>0}$ that satisfy the following:\\
[color=#FFFFFF]___[/color]1. For all sets $A, B \in \mathbb{S}$ we have \[f(A \cap B) = \text{min}(f(A), f(B)).\]
[color=#FFFFFF]___[/color]2. For all positive integers $n$ we have \[\sum \limits_{X \subseteq [1, n]}
f(X) = 2^{n+1}-1.\]
(Here, by a proper subset $X$ of $\mathbb{Z}_{>0}$ we mean $X \subset \mathbb{Z}_{>0}$ with $X \ne \mathbb{Z}_{>0}$. It is allowed for $X$ to have infinite size.) \\
[i]Proposed by MV Adhitya, Kanav Talwar, Siddharth Choppara, Archit Manas[/i]
Find all $f: \mathbb{Q}_{+} \rightarrow \mathbb{R}$ such that \[ f(x)+f(y)+f(z)=1 \] holds for every positive rationals $x, y, z$ satisfying $x+y+z+1=4xyz$.
Let $\mathbb{N}_0=\{0,1,2 \cdots \}$. Does there exist a function $f: \mathbb{N}__0 \to \mathbb{N}_0$ such that:
\[ f^{2003}(n)=5n, \forall n \in \mathbb{N}_0 \]
where we define: $f^1(n)=f(n)$ and $f^{k+1}(n)=f(f^k(n))$, $\forall k \in \mathbb{N}_0$?
Function $f$ is defined on the positive integers by $f(1)=1$, $f(2)=2$ and
$$f(n+2)=f(n+2-f(n+1))+f(n+1-f(n))\enspace\text{for }n\ge1.$$
(a) Prove that $f(n+1)-f(n)\in\{0,1\}$ for each $n\ge1$.
(b) Show that if $f(n)$ is odd then $f(n+1)=f(n)+1$.
(c) For each positive integer $k$ find all $n$ for which $f(n)=2^{k-1}+1$.