Found problems: 2349
Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that
$$f(f(xy-x))+f(x+y)=yf(x)+f(y)$$
for all real numbers $x$ and $y$.
Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f\colon \mathbb{N}\to \mathbb{N}$ such that $mf(m)+(f(f(m))+n)^2$ divides $4m^4+n^2f(f(n))^2$ for all positive integers $m$ and $n$.
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Find all real numbers $a$ such that there exist $f:\mathbb{R} \to \mathbb{R}$ with $$f(x-f(y))=f(x)+a[y]$$ for all $x,y\in \mathbb{R}$
Find all functions $f:\mathbb{R}\to\mathbb{R}$ so that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y$ that belongs to $\mathbb{R}$.
Let $\mathbb{R}$ be the set of all real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying for all $x,y\in\mathbb{R}$ the equation $f(x)+f(y)=f(f(x)f(y))$.
Find all functions $f : \mathbb{Z} \to\mathbb{ Z}$ such that
\[ n^2+4f(n)=f(f(n))^2 \]
for all $n\in \mathbb{Z}$.
[i]Proposed by Sahl Khan, UK[/i]
Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$.
[i]Sammy Luo and Alex Zhu.[/i]
Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties:
(a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$.
(b) We have $f\left(2\right) = 0$.
(c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$.
[b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.
Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ that satisfy the following conditions:
- $f(xyz)+f(x)+f(y)+f(z)=f(\sqrt{xy})f(\sqrt{yz})f(\sqrt{zx})$ for all $x,y,z\in\mathbb{R}^+$;
- $f(x)<f(y)$ for all $1\le x<y$.
[i]Proposed by Hojoo Lee, Korea[/i]
Find all functions $f : R-\{0\} \to R$ which satisfy $(1 + y)f(x) - (1 + x)f(y) = yf(x/y) - xf(y/x)$ for all real $x, y \ne 0$, and which take the values $f(1) = 32$ and $f(-1) = -4$.
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that
\[x+f(yf(x)+1)=xf(x+y)+yf(yf(x))\]
for all $x,y>0.$
Let $\mathbb{Z^+}$ denote the set of positive integers. Find all functions $f:\mathbb{Z^+} \to \mathbb{Z^+}$ satisfying the condition
$$ f(a) + f(b) \mid (a + b)^2$$
for all $a,b \in \mathbb{Z^+}$
Find all functions $f:\mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that for all positive integers $m,n$ and $a$ we have
a) $f(f(m)f(n))=mn$ and
b) $f(2024a+1)=2024a+1$.
Find all monotonic functions $u:\mathbb{R}\rightarrow\mathbb{R}$ which have the property that there exists a strictly monotonic function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x+y)=f(x)u(x)+f(y) \]
for all $x,y\in\mathbb{R}$.
[i]Vasile Pop[/i]
Let $m,n \in Z, m\ne n, m \ne 0, n \ne 0$ .
Find all $f: Z \to R$ such that $f(mx+ny)=mf(x)+nf(y)$ for all $x,y \in Z$ .
Let $P(z)$ and $Q(z)$ be complex-variable polynomials, with degree not less than $1$. Let
\[P_k = \{z \in \mathbb C | P(z) = k \}, Q_k = \{ z \in \mathbb C | Q(z) = k \}.\]
Let also $P_0 = Q_0$ and $P_1 = Q_1$. Prove that $P(z) \equiv Q(z).$
Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ such that for all integers $x$, $y$, $$f(x-f(y))=f(f(y))+f(x-2y)$$
[i]Proposed by Ivan Chan Kai Chin[/i]
Find all functions $f : R \to R$ such that the inequality $$\sum_{i=1}^{2549} f(x_i + x_{i+1}) + f (\sum_{i=1}^{2550}x_y) \le \sum_{i=1}^{2550}f(2x_i)$$ for all reals $x_1, x_2, . . . , x_{2550}$.
Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.
22. Let the function $f : \mathbb{Z} \to \mathbb{Z}$ take only integer inputs and have integer outputs. For any integers $x$ and $y$, f satisfies
$$f(x) + f(y) = f(x + 1) + f(y - 1)$$
If $f(2016) = 6102$ and $f(6102) = 2016$, what is $f(1)$?
23. Let $d$ be a randomly chosen divisor of $2016$. Find the expected value of
$$\frac{d^2}{d^2 + 2016}$$
24. Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjecent vertices consists of edges with at least two different colors. What is the smallest possible value of $N$?
Find all functions $f : \mathbb{R} \mapsto \mathbb{R} $ such that
$$ \left(f(x)+y\right)\left(f(x-y)+1\right)=f\left(f(xf(x+1))-yf(y-1)\right)$$
for all $x,y \in \mathbb{R}$
Let \(\mathbb R_{>0}\) denote the set of positive real numbers. Find all functions \(f:\mathbb R_{>0}\to\mathbb R_{>0}\) such that for all positive real numbers \(x\) and \(y\), \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\]
[i]Proposed by Luke Robitaille[/i]
Find all functions $f : R \to R$ satisfying $f(x + xy + f(y))=(f(x) + \frac12)(f(y) + \frac12 )$ for all $x, y \in R$.