Found problems: 2349
Let the $f:\mathbb{N}\rightarrow\mathbb{N}$ be the function such that
(i) For all positive integers $n,$ $f(n+f(n))=f(n)$
(ii) $f(n_o)=1$ for some $n_0$
Prove that $f(n)\equiv 1.$
Find all real functions $f$ definited on positive integers and satisying:
(a) $f(x+22)=f(x)$,
(b) $f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)$
for all positive integers $x$ and $y$.
We call a function $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ [i]good[/i] if for all $x,y \in \mathbb{Q}^+$ we have: $$f(x)+f(y)\geq 4f(x+y).$$
a) Prove that for all good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ $$f(x)+f(y)+f(z) \geq 8f(x+y+z)$$
b) Does there exists a good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ such that $$f(x)+f(y)+f(z) < 9f(x+y+z) ?$$
Let $f:\mathbb{R}\rightarrow (0,\infty)$ be continuous function of period $1$. Prove that for any $a\in\mathbb{R}$ $$\int_0^1\frac{f(x)}{f(x+a)}dx\geq 1.$$
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$
[i]Proposed by Pierre Bornsztein, France[/i]
Find all polynomials $p$ with real coefficients such that for all real $x$ , $xp(x)p(1-x)+x^3 +100 \ge 0$.
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x, y\in \mathbb{R}$ the following holds:
\[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]
Let $\mathbb{N} = \{1, 2, 3, \dots \}$ be the set of positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for all positive integers $n$ and prime numbers $p$:
$$p \mid f(n)f(p-1)!+n^{f(p)}.$$
[i]Proposed by Dorlir Ahmeti[/i]
a) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+1) - f(n)$ for all $n \in N$?
b) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+2) - f(n)$ for all $n \in N$?
I. Voronovich
Find all positive integers $k$ for which there exists a nonlinear function $f:\mathbb{Z} \rightarrow\mathbb{Z}$ such that the equation $$f(a)+f(b)+f(c)=\frac{f(a-b)+f(b-c)+f(c-a)}{k}$$ holds for any integers $a,b,c$ satisfying $a+b+c=0$ (not necessarily distinct).
[i]Evan Chen[/i]
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
Let $f$ be a function such that $f(x)+f(x+1)=2^x$ and $f(0)=2010$. Find the last two digits of $f(2010)$.
Find all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$
and are strictly monotone in $(0,+\infty )$
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that
\[f(x+y) + f(x)f(y) = f(xy) + (y+1)f(x) + (x+1)f(y)\]
for all $x,y \in \mathbb{R}$.
Let $f: Z^+ \to Z^+ \cup \{0\}$ a function that meets the following conditions:
a) $f (a b) = f (a) + f (b)$,
b) $f (a) = 0$ provided that the digits of the unit of $a$ are $7$,
c) $f (10) = 0$.
Find $f (2016).$
Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \neq 0 = f(1)$. Assume also that $f$ satisfies equations [b](A)[/b] and [b](B)[/b] below. \begin{eqnarray*}f(xy) = f(x) + f(y) -f(x) f(y)\qquad\mathbf{(A)}\\
f(x-y) f(x) f(y) = f(0) f(x) f(y)\qquad\mathbf{(B)}
\end{eqnarray*} for all integers $x,y$.
[b](i)[/b] Determine explicitly the set $\big\{f(a)~:~a\in\mathbb{Z}\big\}$.
[b](ii)[/b] Assuming that there is a non-zero integer $a$ such that $f(a) \neq 0$, prove that the set $\big\{b~:~f(b) \neq 0\big\}$ is infinite.
Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.
Find all real functions $f$ defined on $ \mathbb R$, such that \[f(f(x)+y) = f(f(x)-y)+4f(x)y ,\] for all real numbers $x,y$.
Find all functions \(f:\mathbb{R} \rightarrow \mathbb{R}\) such that
\[ f(x^2+y^2f(x)) = xf(y)^2-f(x)^2 \]
for every \(x, y \in \mathbb{R}\)
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 $f: \mathbb N \to \mathbb N$ for which
\[ f(n) + f(n+1) = f(n+2)f(n+3)-1996\]
holds for all positive integers $n$.
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying
\[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \]
for all positive integer $m,n$.
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(f(x)-f(y))+2f(xy)=x^2f(x)+f(y^2)$$ for all real numbers $x,y$.
$f: \mathbb{R} \rightarrow \mathbb{R}$ is a function such that for $\forall x,y\in \mathbb{R}$ the equation
$f(xy+x+y)=f(xy)+f(x)+f(y)$
is true. Prove that $f(x+y)=f(x)+f(y)$ for $\forall$ $x,y\in \mathbb{R}$.
Show that there exists no function $f : Z \to Z$ such that for all $m, n \in Z$
$$f(m + f(n)) = f(m) - n.$$