Solve for $g : \mathbb{Z}^+ \to \mathbb{Z}^+$ such that $$g(m + n) = g(m) + mn(m + n) + g(n)$$ Show that $g(n)$ is of the form $\sum_{i=0}^{d} {c_i n^i}$ \\ and find necessary and sufficient conditions on $d$ and $c_0, c_1, \cdots , c_d$

Find all functions $f\colon \mathbb{Q}\to \mathbb{R}_{\geq 0}$ such that for any two rational numbers $x$ and $y$ the following conditions hold [list] [*] $f(x+y)\leq f(x)+f(y)$, [*]$f(xy)=f(x)f(y)$, [*]$f(2)=1/2$. [/list]

Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all integers $x$ and $y$, the number $$f(x)^2+2xf(y)+y^2$$ is a perfect square. [i]Proposed by Barış Koyuncu[/i]

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 $n$ be a positive integer. Find all functions $f\colon \mathbb{R}\rightarrow \mathbb{R}$ that satisfy the equation $$ (f(x))^n f(x+y) = (f(x))^{n+1} + x^n f(y) $$ for all $x ,y \in \mathbb{R}$.

Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties: (i) $f(1) = 2$; (ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.

Consider functions $f$ that satisfy $|f(x)-f(y)|\leq \frac{1}{2}|x-y|$ for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, what is the greatest possible value of $$f(f(800))-f(f(400))?$$ $ \textbf{(A)}\ 25 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 100 \qquad \textbf{(D)}\ 150 \qquad \textbf{(E)}\ 200$

Find all functions $f : R \to R$ satisfying $f(x + xy + f(y))=(f(x) + \frac12)(f(y) + \frac12 )$ for all $x, y \in R$.

Find all strictly increasing functions $f : Z \to R$ such that for any $m, n \in Z$ there exists a $k \in Z$ such that $f(k) = f(m) - f(n)$. Nguyễn Duy Thái Sơn

The function $f:\mathbb{R} \to \mathbb{R}$ is such that $f(f(x+1)) = x^3+1$ for all real numbers $x$. Prove that the equation $f(x) = 0 $ has exactly one real root.

Find all functions $f : R \to R$ satisfying $f(xy - 1) + f(x)f(y) = 2xy - 1$ for all real numbers $x, y$

Find all continuous monotonic functions $f : R \to R$ that satisfy $f (1) = 1$ and $f(f (x)) = f (x)^2$ for all $x \in R$.

Find all pairs of functions $f; g : R \to R$ such that for all reals $x.y \ne 0$ : $$f(x + y) = g \left(\frac{1}{x}+\frac{1}{y}\right) \cdot (xy)^{2008}$$

Let $\mathbb{R}$ denote the set of all real numbers. Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$, \[f(xy+xf(x))=f(x)\left(f(x)+f(y)\right).\]

For each integer $k\geq 2$, determine all infinite sequences of positive integers $a_1$, $a_2$, $\ldots$ for which there exists a polynomial $P$ of the form \[ P(x)=x^k+c_{k-1}x^{k-1}+\dots + c_1 x+c_0, \] where $c_0$, $c_1$, \dots, $c_{k-1}$ are non-negative integers, such that \[ P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k} \] for every integer $n\geq 1$.

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that \[f(x(x + f(y))) = (x + y)f(x),\] for all $x, y \in\mathbb{R}$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)+f(x+y)=xy$ for all real numbers $x$ and $y$.

$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.

In the Cartesian plane $\mathbb{R}^2,$ each triangle contains a Mediterranean point on its sides or in its interior, even if the triangle is degenerated into a segment or a point. The Mediterranean points have the following properties: [b](i)[/b] If a triangle is symmetric with respect to a line which passes through the origin $(0,0)$, then the Mediterranean point lies on this line. [b](ii)[/b] If the triangle $DEF$ contains the triangle $ABC$ and if the triangle $ABC$ contains the Mediterranean points $M$ of $DEF,$ then $M$ is the Mediterranean point of the triangle $ABC.$ Find all possible positions for the Mediterranean point of the triangle with vertices $(-3,5),\ (12,5),\ (3,11).$

Let $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}$ be a function with the following properties: (i) $f(1) = 0$ (ii) $f(p) = 1$ for all prime numbers $p$ (iii) $f(xy) = y \cdot f(x) + x \cdot f(y)$ for all $x,y$ in $\mathbb{Z}_{>0}$ Determine the smallest integer $n \ge 2015$ that satisfies $f(n) = n$. (Gerhard J. Woeginger)

Denote the set of positive rational numbers by $\mathbb{Q}_{+}$. Find all functions $f: \mathbb{Q}_{+}\rightarrow \mathbb{Q}_{+}$ that satisfy $$\underbrace{f(f(f(\dots f(f}_{n}(q))\dots )))=f(nq)$$ for all integers $n\ge 1$ and rational numbers $q>0$.

Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, such that 1) $f(0,x)$ is non-decreasing ; 2) for any $x,y \in \mathbb{R}$, $f(x,y)=f(y,x)$ ; 3) for any $x,y,z \in \mathbb{R}$, $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ; 4) for any $x,y,a \in \mathbb{R}$, $f(x+a,y+a)=f(x,y)+a$ .

Let $f:\mathbb{N}\rightarrow \mathbb{N}$ be a strictly increasing function such that $f(2)=2$ and $f(mn)=f(m)f(n)$ for every pair of relatively prime positive integers $m$ and $n$. Prove that $f(n)=n$ for every positive integer $n$.

For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.

Determine is there a function $a: \mathbb{N} \rightarrow \mathbb{N}$ such that: $i)$ $a(0)=0$ $ii)$ $a(n)=n-a(a(n))$, $\forall n \in$ $ \mathbb{N}$. If exists prove: $a)$ $a(k)\geq a(k-1)$ $b)$ Does not exist positive integer $k$ such that $a(k-1)=a(k)=a(k+1)$.