Found problems: 1513
Show that there is no infinite sequence an of natural numbers such that \[a_{a_n}=a_{n+1}a_{n-1}-a_{n}^2\] for all $n\geq 2$
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y,z$, we have $$(f(x)+1)(f(y)+f(z))=f(xy+z)+f(xz-y)$$
Determine all strictly increasing functions $f : R \to R$ such that for all $x \ne y$ to hold $$\frac{2\left[f(y)-f\left(\frac{x+y}{2}\right) \right]}{f(x)-f(y)}=\frac{f(x)-f(y)}{2\left[f\left(\frac{x+y}{2}\right)-f(x) \right]}$$
Determine all functions $f : N_0 \to R$ satisfying $f (x+y)+ f (x-y)= f (3x)$ for all $x,y$.
Let $S$ be a set of positive integers $n_1, n_2, \cdots, n_6$ and let $n(f)$ denote the number $n_1n_{f(1)} +n_2n_{f(2)} +\cdots+n_6n_{f(6)}$, where $f$ is a permutation of $\{1, 2, . . . , 6\}$. Let
\[\Omega=\{n(f) | f \text{ is a permutation of } \{1, 2, . . . , 6\} \} \]
Give an example of positive integers $n_1, \cdots, n_6$ such that $\Omega$ contains as many elements as possible and determine the number of elements of $\Omega$.
Determine all functions $f : \mathbb {N} \rightarrow \mathbb {N}$ such that $f$ is increasing (not necessarily strictly) and the numbers $f(n)+n+1$ and $f(f(n))-f(n)$ are both perfect squares for every positive integer $n$.
Let $f:\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ be a continuous function such that $f(0)=0$ and $$f(x)+f(f(x))+f(f(f(x)))=3x$$ for all $x>0$. Show that $f(x)=x$ for all $x>0$.
Let be $f: (0,+\infty)\rightarrow \mathbb{R}$ monoton-decreasing .
If $f(2a^2+a+1)<f(3a^2-4a+1)$ find interval of $a$ .
Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying
(i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$.
(ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
Find all non-decreasing functions from real numbers to itself such that for all real numbers $x,y$ $f(f(x^2)+y+f(y))=x^2+2f(y)$ holds.
Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that
\[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\]
for all $x,y \in \mathbb{R} $.
Let $f : Q \to Q$ be a function satisfying the equation $f(x + y) = f(x) + f(y) + 2547$ for all rational numbers $x, y$. If $f(2004) = 2547$, find $f(2547)$.
Determine all Functions $f:\mathbb{Z} \to \mathbb{Z}$ such that $f(f(a)-b)+bf(2a)$ is a perfect square for all integers $a$ and $b$.
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y\in(1,\infty)$ and $x^2\le y\le x^3,$ then $(f(x))^2\le f(y) \le (f(x))^3.$
Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$.
Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}^{+}$ such that for all positive real numbers $x$ and $y$, the following equation holds:
\[(x+y)f(f(x)y)=x^2f(f(x)+f(y)).\]
Find all functions $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $m,n\in \mathbb{N}_{0}$: \[mf(n)+nf(m)=(m+n)f(m^{2}+n^{2}).\]
Let $\mathbb{R}^+$ denote the set of positive real numbers. Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all positive real numbers $x$ and $y$ : \[f(x)f(y+f(x))=f(1+xy)\]
[i]Proposed by Otgonbayar Uuye. [/i]
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(xy+f(x))+f(y)=xf(y)+f(x+y),\]for all real numbers $x,y$.
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that\[f\left(\Big \lceil \frac{f(m)}{n} \Big \rceil\right)=\Big \lceil \frac{m}{f(n)} \Big \rceil\]for all $m,n \in \mathbb{N}$.
[i]Proposed by Md. Ashraful Islam Fahim[/i]
Prove that there is a function $ F:\mathbb{N}\longrightarrow\mathbb{N} $ satisfying $ (F\circ F) (n) =n^2, $ for all $ n\in\mathbb{N} . $
The function $f$ is defined for non-negative integers and satisfies the condition $f(n) = f(f(n + 11))$, if $n \le 1999$ and
$f(n) = n - 5$, if $n > 1999$. Find all solutions of the equation $f(n) = 1999$.
Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies
(i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$.
(ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.