Found problems: 2349
Find all function $f:\mathbb{Z}\to\mathbb{Z}$ satisfying
$\text{(i)}$ $f(f(m)+n)+2m=f(n)+f(3m)$ for every $m,n\in\mathbb{Z}$,
$\text{(ii)}$ there exists a $d\in\mathbb{Z}$ such that $f(d)-f(0)=2$, and
$\text{(iii)}$ $f(1)-f(0)$ is even.
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
$$(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$$
Consider a function $f: R \to R$ that fulfills the following two properties: $f$ is periodic of period $5$ (that is, for all $x\in R$, $f (x + 5) = f (x)$), and by restricting $f$ to the interval $[-2,3]$, $f$ coincides to $x^2$. Determine the value of $f(2018).$
Find all functions $f : R \to R$ such that $x^2f(yf(x))= y^2f(x)f(f(x))$ for all real numbers $x$ and $y$.
Let $f : N \to N$ be a function which satisfies $f(x)+ f(x+2) \le 2 f(x+1)$ for any $x \in N$.
Prove that there exists a line in the coordinate plane containing infinitely many points of the form $(n, f(n)), n \in N$.
Find all functions $f : R -\{0\} \to R$ that satisfy $\frac{1}{x}f(-x)+ f\left(\frac{1}{x}\right)= x$ for all $x \ne 0$.
Find all injective functions $f : R \to R$ such that for all real $x \ne y$ , $f\left(\frac{x+y}{x-y}\right) = \frac{f(x)+ f(y)}{f(x)- f(y)}$
Find all functions $f : R \to R$ that satisfy the functional equation $$f(x + f(xy)) = f(x)(1 + y).$$
Let $f$ be a non-zero function from the set of positive integers to the set of non-negative integers such that for all positive integers $a$ and $b$ we have
$$2f(ab)=(b+1)f(a)+(a+1)f(b).$$
Prove that for every prime number $p$ there exists a prime $q$ and positive integers $x_{1}$, ..., $x_{n}$ and $m \geq 0$ so that
$$\frac{f(q^{p})}{f(q)} = (px_{1}+1) \cdot ... \cdot (px_{n}+1) \cdot p^{m},$$
where the integers $px_{1}+1$,..., $px_{n}+1$ are all prime.
[i]Authored by Nikola Velov[/i]
The function f is defined on the set of integers and satisfies
$\bullet$ $f(n) = n - 2$, if $n \ge 2005$
$\bullet$ $f(n) = f(f(n+7))$, if $n < 2005$.
Find $f(3)$.
Let \(c\) be a real number. For all \(x\) and \(y\) real numbers we have,
\[f(x-f(y))=f(x-y)+c(f(x)-f(y))\]
and \(f(x)\) is not constant.
\(a)\) Find all possible values of \(c\).
\(b)\) Can \(f\) be periodic?
Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths
\[ a, f(b) \text{ and } f(b \plus{} f(a) \minus{} 1).\]
(A triangle is non-degenerate if its vertices are not collinear.)
[i]Proposed by Bruno Le Floch, France[/i]
$f(x)$ is a real valued function defined for $x \geq 0$ such that $f(0) = 0$, $f(x+1)=f(x)+\sqrt{x}$ for all $x$, and
\[
f(x) < \frac{1}{2}f\left(x - \frac{1}{2}\right)+\frac{1}{2}f\left(x + \frac{1}{2}\right) \quad \text{for all} \quad x \geq \frac{1}{2}
\]
Show that $f\left(\frac{1}{2}\right)$ is uniquely determined.
Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation
\[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\]
holds.
Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.
Find all functions $f : Z_{>0} \to Z_{>0}$ for which $f(n) | f(m) - n$ if and only if $n | m$ for all natural numbers $m$ and $n$.
Find all functions $f : R \to R$ such that
$$f(f(x) + f(y)^2) = f(x)^2 +y^2f(y)^3.$$
Here $R$ denotes the usual real numbers.
Determine all functions $f:\mathbb{R}\to\mathbb{Z}$ satisfying the inequality $(f(x))^2+(f(y))^2 \leq 2f(xy)$ for all reals $x,y$.
Let $S$ be the set of all pairs $(m,n)$ of relatively prime positive integers $m,n$ with $n$ even and $m < n.$ For $s = (m,n) \in S$ write $n = 2^k \cdot n_o$ where $k, n_0$ are positive integers with $n_0$ odd and define \[ f(s) = (n_0, m + n - n_0). \] Prove that $f$ is a function from $S$ to $S$ and that for each $s = (m,n) \in S,$ there exists a positive integer $t \leq \frac{m+n+1}{4}$ such that \[ f^t(s) = s, \] where \[ f^t(s) = \underbrace{ (f \circ f \circ \cdots \circ f) }_{t \text{ times}}(s). \]
If $m+n$ is a prime number which does not divide $2^k - 1$ for $k = 1,2, \ldots, m+n-2,$ prove that the smallest value $t$ which satisfies the above conditions is $\left [\frac{m+n+1}{4} \right ]$ where $\left[ x \right]$ denotes the greatest integer $\leq x.$
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$:
(i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$
(ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$
[i]Proposed by Ashwin Sah[/i]
Find all functions $f : \mathbb{Z}\to \mathbb{Z}$ that satisfy:
$i) f(f(x))=x, \forall x\in\mathbb{Z}$
$ii)$ For any integer $x$ and $y$ such that $x + y$ is odd, it holds that $f(x) + f(y) \ge x + y.$
Find all $f\colon\mathbf{R}\rightarrow\mathbf{R}$ such that
\[f(f(x)f(y)) = f(x + y) + f(xy)\]
for all $x,y\in\mathbf{R}$.
Find all $a\in\mathbb{N}$ such that exists a bijective function $g :\mathbb{N} \to \mathbb{N}$ and a function $f:\mathbb{N}\to\mathbb{N}$, such that for all $x\in\mathbb{N}$,
$$f(f(f(...f(x)))...)=g(x)+a$$ where $f$ appears $2009$ times.
[i](tatari/nightmare)[/i]
Find all functions $ f :\mathbb{R}_{0}^{+}\mapsto\mathbb{R}_{0}^{+} $ satisfying the conditions $4f(x)\geq 3x$ and $f(4f(x)-3x)=x$ for all $x\geq 0$ .
Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose
\[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\]
Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$