Find all functions $f:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ such that $x,y\in \mathbb{R}^{+},$ \[ f\left(\frac{y}{f(x+1)}\right)+f\left(\frac{x+1}{xf(y)}\right)=f(y) \]

Let be two nonzero real numbers $ a,b, $ and a function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ satisfying the functional equation $$ f(x+a+b)+f(x)=f(x+a)+f(x+b) . $$ [b]1)[/b] Prove that $ f $ is periodic if $ a/b $ is rational. [b]2)[/b] If $ a/b $ is not rational, could $ f $ be nonperiodic?

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for any pair of naturals $m,n$, $$\gcd(f(m),n) = \gcd(m,f(n)).$$

Let $y=f(x)$ be a function of the graph of broken line connected by points $(-1,\ 0),\ (0,\ 1),\ (1,\ 4)$ in the $x$ -$y$ plane. Find the minimum value of $\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.$ [i]2010 Tohoku University entrance exam/Economics, 2nd exam[/i]

We say that the real-valued function $ f(x)$ defined on the interval $ (0,1)$ is approximately continuous on $ (0,1)$ if for any $ x_0 \in (0,1)$ and $ \varepsilon >0$ the point $ x_0$ is a point of interior density $ 1$ of the set \[ H\equal{} \{x : \;|f(x)\minus{}f(x_0)|< \varepsilon \ \}.\] Let $ F \subset (0,1)$ be a countable closed set, and $ g(x)$ a real-valued function defined on $ F$. Prove the existence of an approximately continuous function $ f(x)$ defined on $ (0,1)$ such that \[ f(x)\equal{}g(x) \;\textrm{for all}\ \;x \in F\ .\] [i]M. Laczkovich, Gy. Petruska[/i]

Deos there exist a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$, $f(x^2y+f(x+y^2))=x^3+y^3+f(xy)$

A real continuous function has the property that its evaluation at any point is nilpotent under composition with itself. Prove that this function is $ 0. $ [i]Vasile Pop[/i]

Let $ f:\mathbb{D}\longrightarrow\mathbb{C} $ be a continuous function, where $ \mathbb{D} $ is the closed unit disk. Suppose that $ f $ is holomorphic on the open unit disk and that $ e^{i\theta } $ are roots, for any $ \theta\in\left[ 0,\pi /4 \right] . $ Show that $ f=0_{\mathbb{D}} . $

Let $f$ be a function for which $f\left(\frac x3\right)=x^2+x+1$. Find the sum of all values of $z$ for which $f(3z)=7$. $\text{(A)}\ -\frac13\qquad\text{(B)}\ -\frac19 \qquad\text{(C)}\ 0 \qquad\text{(D)}\ \frac59 \qquad\text{(E)}\ \frac53$

Let $A$ be the set of all polynomials $f(x)$ of order $3$ with integer coefficients and cubic coefficient $1$, so that for every $f(x)$ there exists a prime number $p$ which does not divide $2004$ and a number $q$ which is coprime to $p$ and $2004$, so that $f(p)=2004$ and $f(q)=0$. Prove that there exists a infinite subset $B\subset A$, so that the function graphs of the members of $B$ are identical except of translations

Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.

Consider the function $ f: \mathbb{N}_0\to\mathbb{N}_0$, where $ \mathbb{N}_0$ is the set of all non-negative integers, defined by the following conditions : $ (i)$ $ f(0) \equal{} 0$; $ (ii)$ $ f(2n) \equal{} 2f(n)$ and $ (iii)$ $ f(2n \plus{} 1) \equal{} n \plus{} 2f(n)$ for all $ n\geq 0$. $ (a)$ Determine the three sets $ L \equal{} \{ n | f(n) < f(n \plus{} 1) \}$, $ E \equal{} \{n | f(n) \equal{} f(n \plus{} 1) \}$, and $ G \equal{} \{n | f(n) > f(n \plus{} 1) \}$. $ (b)$ For each $ k \geq 0$, find a formula for $ a_k \equal{} \max\{f(n) : 0 \leq n \leq 2^k\}$ in terms of $ k$.

Determine all functions $f: \mathbb{Z}_{\geq 2025} \to \mathbb{Z}_{>0}$ such that $mn+1$ divides $f(m)f(n) + 1$ for any integers $m,n \geq 2025$ and there exists a polynomial $P$ with integer coefficients, such that $f(n) \leq P(n)$ for all $n\geq 2025$.

Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(xf(y)+y)+f(-f(x))=f(yf(x)-y)+y\] for all $x,y\in\mathbb{R}$

Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying \[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\ 1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\ + f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\ + f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases} \] for all nonnegative integers $ p$, $ q$, $ r$.

An isosceles trapezoid has legs and shorter base of length $1$. Find the maximum possible value of its area

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that\[f(x)+f(t)=f(y)+f(z)\]for all rational numbers $x<y<z<t$ that form an arithmetic progression. ($\mathbb{Q}$ is the set of all rational numbers.)

Let $ X $ be the power set of set of $ \{ 0\}\cup\mathbb{N} , $ and let be a function $ d:X^2\longrightarrow\mathbb{R} $ defined as $$ d(U,V)=\sum_{n\in\mathbb{N}}\frac{\chi_U (n) +\chi_V (n) -2\chi_{U\cap V} (n)}{2} , $$ where $ \chi_W (n)=\left\{ \begin{matrix} 1,& n\in W\\ 0,& n\not\in W \end{matrix} \right. ,\quad\forall W\in X,\forall n\in\mathbb{N} . $ [b]a)[/b] Prove that there exists an unique $ V' $ such that $ \lim_{k\to\infty} d\left( \{ k+i|i\in\mathbb{N}\} , V'\right) =0. $ [b]b)[/b] Demonstrate that for all $ V\in X $ there exists a $ v\in\mathbb{N} $ with $ d\left( \left\{ \frac{3}{2} -\frac{1}{2}(-1)^{v} \right\} , V \right) >\frac{1}{k} . $ [b]c)[/b] Let $ f: X\longrightarrow X,\quad f(X)=\left\{ 1+x|x\in X\right\} . $ Calculate $ d\left( f(A),f(B) \right) $ in terms of $ d(A,B) $ and prove that $ f $ admits an unique fixed point.

Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\] [i]Proposed by Japan[/i]

Three points $A,B,C$ are such that $B\in AC$. On one side of $AC$, draw the three semicircles with diameters $AB,BC,CA$. The common interior tangent at $B$ to the first two semicircles meets the third circle $E$. Let $U,V$ be the points of contact of the common exterior tangent to the first two semicircles. Evaluate the ratio $R=\frac{[EUV]}{[EAC]}$ as a function of $r_{1} = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$, where $[X]$ denotes the area of polygon $X$.

[color=darkred]Let $g:\mathbb{R}\to\mathbb{R}$ be a continuous and strictly decreasing function with $g(\mathbb{R})=(-\infty,0)$ . Prove that there are no continuous functions $f:\mathbb{R}\to\mathbb{R}$ with the property that there exists a natural number $k\ge 2$ so that : $\underbrace{f\circ f\circ\ldots\circ f}_{k\text{ times}}=g$ . [/color]

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

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]

$$\left\lfloor\left(1\cdot2+2\cdot2^2+\ldots+100\cdot2^{100}\right)\cdot9^{-901}\right\rfloor=?$$

$d(n)$ shows the number of positive integer divisors of positive integer $n$. For which positive integers $n$ one cannot find a positive integer $k$ such that $\underbrace{d(\dots d(d}_{k\ \text{times}} (n) \dots )$ is a perfect square.