Suppose that $T\in \mathbb N$ is given. Find all functions $f:\mathbb Z \to \mathbb C$ such that, for all $m\in \mathbb Z$ we have $f(m+T)=f(m)$ and: $$\forall a,b,c \in \mathbb Z: f(a)\overline{f(a+b)f(a+c)}f(a+b+c)=1.$$ Where $\overline{a}$ is the complex conjugate of $a$.

Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies (i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$. (ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.

The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$

Let \( \mathbb{R} \) be the set of real numbers. Determine all functions \( f: \mathbb{R} \to \mathbb{R} \) such that, for any real numbers \( x \) and \( y \), \[ f(x^2 y - y) = f(x)^2 f(y) + f(x)^2 - 1. \]

Find all functions $f,g: \mathbb{Q}\to \mathbb{Q}$ such that the following two conditions hold: $$f(g(x)-g(y))=f(g(x))-y \ \ (1)$$ $$g(f(x)-f(y))=g(f(x))-y\ \ (2)$$ for all $x,y \in \mathbb{Q}$.

Let $\mathbb{R}^{\ge 0}$ be the set of all nonnegative real numbers. Find all functions $f:\mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that $$ x+2 \max\{y,f(x),f(z)\} \ge f(f(x))+2 \max\{z,f(y)\}$$ for all nonnegative real numbers $x,y$ and $z$.

Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$

We call a function $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ [i]good[/i] if for all $x,y \in \mathbb{Q}^+$ we have: $$f(x)+f(y)\geq 4f(x+y).$$ a) Prove that for all good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ $$f(x)+f(y)+f(z) \geq 8f(x+y+z)$$ b) Does there exists a good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ such that $$f(x)+f(y)+f(z) < 9f(x+y+z) ?$$

Find all functions \(f:\mathbb{R}^{+} \to \mathbb{R}^{+}\) such that \[ f(x) > x \ \ \text{and} \ \ f(x-y+xy+f(y)) = f(x+y) + xf(y) \] for arbitrary positive real numbers \(x\) and \(y\).

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

It is given function $f : A \rightarrow \mathbb{R}$, $(A\subseteq \mathbb{R})$ such that $$f(x+y)=f(x)\cdot f(y)-f(xy)+1; (\forall x,y \in A)$$ If $f : A \rightarrow \mathbb{R}$, $(\mathbb{N} \subseteq A\subseteq \mathbb{R})$ is solution of given functional equation, prove that: $$f(n)=\begin{cases} \frac{c^{n+1}-1}{c-1} \text{, } \forall n \in \mathbb{N}, c \neq 1 \\ n+1 \text{, } \forall n \in \mathbb{N}, c = 1 \end{cases}$$ where $c=f(1)-1$ $a)$ Solve given functional equation for $A=\mathbb{N}$ $b)$ With $A=\mathbb{Q}$, find all functions $f$ which are solutions of the given functional equation and also $f(1997) \neq f(1998)$

Let $\alpha\in\mathbb{Q}^+$. Determine all functions $f:\mathbb{Q}^+\to\mathbb{Q}^+$ that for all $x,y\in\mathbb{Q}^+$ satisfy the equation \[ f\left(\frac{x}{y}+y\right) ~=~ \frac{f(x)}{f(y)}+f(y)+\alpha x.\] Here $\mathbb{Q}^+$ denote the set of positive rational numbers. (Proposed by Walther Janous)

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 $(-3,5),\ (12,5),\ (3,11).$

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

Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.

For some positive integer $n,$ Elmo writes down the equation \[x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.\] Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation \[f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.\] Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$. Is it possible that there is no integer $k$ (possibly depending on $f$) such that $f^k(x)=x$ for all $x$? [i]Srinivas Arun[/i]

Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$, $$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$(a) Calculate $f(k)$ for $0\le k\le12$. (b) Calculate $f(n)$ for any natural number $n$.

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that if $a,b,c$ are the length sides of a triangle, and $r$ is the radius of its incircle, then $f(a),f(b),f(c)$ also form a triangle where its radius of the incircle is $f(r)$.

$f$ 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 $x$.

Find all functions $f$ mapping non-empty finite sets of integers, to integers, such that $$f(A+B)=f(A)+f(B)$$ for all non-empty sets of integers $A$ and $B$. $A+B$ is defined as $\{a+b: a \in A, b \in B\}$.

Let $\mathbb Z$ be the set of integers. We consider functions $f :\mathbb Z\to\mathbb Z$ satisfying \[f\left(f(x+y)+y\right)=f\left(f(x)+y\right)\] for all integers $x$ and $y$. For such a function, we say that an integer $v$ is [i]f-rare[/i] if the set \[X_v=\{x\in\mathbb Z:f(x)=v\}\] is finite and nonempty. (a) Prove that there exists such a function $f$ for which there is an $f$-rare integer. (b) Prove that no such function $f$ can have more than one $f$-rare integer. [i]Netherlands[/i]

Find all function $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for every three real positive number $x,y,z$ : $$ x+f(y) , f(f(y)) + z , f(f(z))+f(x) $$ are length of three sides of a triangle and for every postive number $p$ , there is a triangle with these sides and perimeter $p$. [i]Proposed by Amirhossein Zolfaghari [/i]

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy $$(f(a)-f(b))(f(b)-f(c))(f(c)-f(a)) = f(ab^2+bc^2+ca^2) - f(a^2b+b^2c+c^2a)$$for all real numbers $a$, $b$, $c$. [i]Proposed by Ankan Bhattacharya, USA[/i]