Found problems: 2349
Find all functions $f: R \to R$ that satisfy $f (x) + 2f (\sqrt[3]{1-x^3}) = x^3$ for all real $x$.
(Here $\sqrt[3]{x}$ is defined all over $R$.)
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $f(a)+f(b)+ab \mid a^2f(a)+b^2f(b)+f(a)f(b)$ for all positive integers $a,b$.
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that for all $x,y\in\mathbb{N}$:
$$0\le y+f(x)-f^{f(y)}(x)\le1$$
that here
$$f^n(x)=\underbrace{f(f(\ldots(f}_{n}(x))\ldots)$$
Let $p = 2017$ be a prime and $\mathbb{F}_p$ be the integers modulo $p$. A function $f: \mathbb{Z}\rightarrow\mathbb{F}_p$ is called [i]good[/i] if there is $\alpha\in\mathbb{F}_p$ with $\alpha\not\equiv 0\pmod{p}$ such that
\[f(x)f(y) = f(x + y) + \alpha^y f(x - y)\pmod{p}\]
for all $x, y\in\mathbb{Z}$. How many good functions are there that are periodic with minimal period $2016$?
[i]Ashwin Sah[/i]
For an integer $ m$, denote by $ t(m)$ the unique number in $ \{1, 2, 3\}$ such that $ m \plus{} t(m)$ is a multiple of $ 3$. A function $ f: \mathbb{Z}\to\mathbb{Z}$ satisfies $ f( \minus{} 1) \equal{} 0$, $ f(0) \equal{} 1$, $ f(1) \equal{} \minus{} 1$ and $ f\left(2^{n} \plus{} m\right) \equal{} f\left(2^n \minus{} t(m)\right) \minus{} f(m)$ for all integers $ m$, $ n\ge 0$ with $ 2^n > m$. Prove that $ f(3p)\ge 0$ holds for all integers $ p\ge 0$.
[i]Proposed by Gerhard Woeginger, Austria[/i]
Find all polynomial function $P$ of real coefficients such that for all $x \in \mathbb{R}$ $$P(x)P(x+1)=P(x^2+2)$$
Consider the function $f : R^+ \to R^+$ and satisfying
$$f(x + 2y + f(x + y)) = f(2x) + f(3y), \,\, \forall \,\, x, y > 0.$$
1. Find all functions $f(x)$ that satisfy the given condition.
2. Suppose that $f(4\sin^4x)f(4\cos^4x) \ge f^2(1)$ for all $x \in \left(0\frac{\pi}{2}\right) $. Find the minimum value of $f(2022)$.
Let $f(n)$ be defined on the positive integers and satisfy:
$f(prime) = 1$, $f(ab) = a f(b) + f(a) b$.
Show that $f$ is unique and find all $n$ such that $n = f(n)$.
For which $a$, there is a function $f: R \to R$, different from a constant, such that
$$f(a(x + y)) = f(x) + f(y) ?$$
Find all functions $f : R_{>0} \to R_{>0}$ with the following property:
$$f \left( \frac{x}{y + 1}\right) = 1 - xf(x + y)$$ for all $x > y > 0$ .
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that for all $x, y\in\mathbb{R}^+$,
\[ \frac{f(x)}{y^2} - \frac{f(y)}{x^2} \le \left(\frac{1}{x}-\frac{1}{y}\right)^2\]
($\mathbb{R}^+$ denotes the set of positive real numbers.)
[i](Proposed by Ivan Chan Guan Yu)[/i]
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$,
\[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]
Find all functions $f : Q \to Q$, such that $$f(x + f (y + f(z))) = y + f(x + z)$$ for all $x ,y ,z \in Q$ .
(I. Voronovich)
Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$.
Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.
Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying: $ x^2 f(x)\plus{}f(1\minus{}x)\equal{}2x\minus{}x^4$ for all $ x \in \mathbb{R}$.
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that we have $f(x + y) = f(x) + f(y) + f(xy)$ for all $ x,y\in \mathbb{R}$
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
Find all functions $f : R- \{-1\} \to R$ such that
$$f(x)f \left( f \left(\frac{1 - y}{1 + y} \right)\right) = f\left(\frac{x + y}{xy + 1}\right) $$
for all $x, y \in R$ that satisfy $(x + 1)(y + 1)(xy + 1) \ne 0$.
Find all the functions $f:\mathbb{N}\to\mathbb{R}$ that satisfy
\[ f(x+y)=f(x)+f(y) \] for all $x,y\in\mathbb{N}$ satisfying $10^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}$.
Note: $\mathbb{N}$ denotes the set of positive integers and $\mathbb{R}$ denotes the set of real numbers.
Let $f(x)$ be the polynomial with integer coefficients ($f(x)$ is not constant) such that
\[(x^3+4x^2+4x+3)f(x)=(x^3-2x^2+2x-1)f(x+1)\]
Prove that for each positive integer $n\geq8$, $f(n)$ has at least five distinct prime divisors.
Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that
$$f(x + f(y)) = f(x) + f(y)$$
for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.
Find all continuous functions $f$ such that $f(x)+ f(x^2) = 0$ for all real numbers $x$.