Found problems: 1513
For which integers $k$ does there exist a function $f : N \to Z$ such that
$f(1995) =1996$ and $f(xy) = f(x)+ f(y)+k f(gcd(x,y))$ for all $x,y \in N$?
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$:
$(i)$ $f(x)=-f(-x);$
$(ii)$ $f(x+1)=f(x)+1;$
$(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x(x + f(y))) = (x + y)f(x),\]
for all $x, y \in\mathbb{R}$.
For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions:
$(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$.
$(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$.
Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.
Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$
Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that
$$|f(A)\cap f(B)|=|A\cap B|$$
whenever $A$ and $B$ are two distinct subsets of $X$.
[i] (Sergiu Novac)[/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] $P(1,0) =1.$
Find all functions $f : \mathbb N \mapsto \mathbb N$ such that the following identity
$$f^{x+1}(y)+f^{y+1}(x)=2f(x+y)$$
holds for all $x,y \in \mathbb N$
Let $A$ be a set of functions $f : R\to R$.
For all $f_1, f_2 \in A$ there exists a $f_3 \in A$ such that $f_1(f_2(y) - x)+ 2x = f_3(x + y)$ for all $x, y \in R$.
Prove that for all $f \in A$, we have $f(x - f(x))= 0$ for all $x \in R$.
Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.
Indicate (justifying your answer) if there exists a function $f: R \to R$ such that for all $x \in R$ fulfills that
i) $\{f(x))\} \sin^2 x + \{x\} cos (f(x)) cosx =f (x)$
ii) $f (f(x)) = f(x)$
where $\{m\}$ denotes the fractional part of $m$. That is, $\{2.657\} = 0.657$, and $\{-1.75\} = 0.25$.
Let $\mathbb{Q}^{+}$ denote the set of positive rational numbers. Find, with proof, all functions $f:\mathbb{Q}^+ \to \mathbb{Q}^+$ such that, for all positive rational numbers $x$ and $y,$ we have \[f(x)=f(x+y)+f(x+x^2f(y)).\]
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, such that for arbitrary $x,y \in \mathbb{R}$: \[ (y+1)f(x)+f(xf(y)+f(x+y))=y.\]
Find all functions $f:\mathbb N\to\mathbb N$ so that
$$f^{2f(b)}(2a)=f(f(a+b))+a+b$$
holds for all positive integers $a,b$.
Find all polynomials $p(x)$ satisfying the condition:
$$p(x^2-2x)=p(x-2)^2.$$
Find all functions $ f : \mathbb{R}\to\mathbb{R}$ that satisfy
\[ f (x^{3} \plus{} y^{3}) \equal{} x^{2}f (x) \plus{} yf (y^{2})
\]
for all $ x, y \in\mathbb R.$
An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions
$f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$).
A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $(f(x + y))^2 = f(x^2) + f(y^2)$ for all $x, y \in \mathbb{Z}$.
Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m,n\in \mathbb{Z}$: \[f(m+f(n)) = f(m)+n.\]
Find all functions $f$ from integers to integers such that
\[ f(m+n) + f(m-n) - 2f(m) = 6mn^2\]
for all integers $m$ and $n$.
Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have
\[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \]
Find the number of [i]good[/i] functions.
Let $\mathbb{N}=\{1, 2, 3, \dots\}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold:
(1) $f(ab) = f(a)f(b)$, and
(2) at least two of the numbers $f(a)$, $f(b)$, and $f(a+b)$ are equal.
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$:
$$f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2$$
[i]Proposed by Ali Behrouz - Mojtaba Zare Bidaki[/i]
Find all functions \( f: \mathbb{R}^{+} \to \mathbb{R}^{+} \) such that:
\[
f(x^2 + y) = xf(x) + \frac{f(y^2)}{y}
\]
for any positive real numbers \( x \) and \( y \).