The function ƒ. which is defined for all real numbers satisfies: $$f(x+y)+f(x-y)=2f(x)+2f(y)$$ Prove that $f(0) = 0$, $f(-x) = f(x)$, $f(2x) = 4 f (x)$, $$f(x + y + z) = f(x + y) + f(y + z) + f(z + x) -f(x) - f(y) -f(z).$$

Determine all functions $ f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfy: $ f(x\plus{}f(y))\equal{}f(x)\plus{}y$ for all $ x,y \in \mathbb{N}$.

Find all functions $f$ of two variables, whose arguments $x,y$ and values $f(x,y)$ are positive integers, satisfying the following conditions (for all positive integers $x$ and $y$): \begin{align*} f(x,x)& =x,\\ f(x,y)& =f(y,x),\\ (x+y)f(x,y)& =yf(x,x+y).\end{align*}

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, \[f(x+yf(x))+y = xy + f(x+y).\] [i]Proposed by Giannis Galamatis, Greece[/i]

Find all functions $f :\mathbb R \to \mathbb R$ such that for all real $x, y$ \[f(f(x)^2 + f(y)) = xf(x) + y.\]

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for any real $x, y$ holds equality $$f(xf(y)) + f(xy) = 2f(x)y$$ [i]Proposed by Arseniy Nikolaev[/i]

[b]p1.[/b] A teenager coining home after midnight heard the hall clock striking the hour. At some moment between $15$ and $20$ minutes later, the minute hand hid the hour hand. To the nearest second, what time was it then? [b]p2.[/b] The ratio of two positive integers $a$ and $b$ is $2/7$, and their sum is a four digit number which is a perfect cube. Find all such integer pairs. [b]p3.[/b] Given the integers $1, 2 , ..., n$ , how many distinct numbers are of the form $\sum_{k=1}^n( \pm k) $ , where the sign ($\pm$) may be chosen as desired? Express answer as a function of $n$. For example, if $n = 5$ , then we may form numbers: $ 1 + 2 + 3- 4 + 5 = 7$ $-1 + 2 - 3- 4 + 5 = -1$ $1 + 2 + 3 + 4 + 5 = 15$ , etc. [b]p4.[/b] $\overline{DE}$ is a common external tangent to two intersecting circles with centers at $O$ and $O'$. Prove that the lines $AD$ and $BE$ are perpendicular. [img]https://cdn.artofproblemsolving.com/attachments/1/f/40ffc1bdf63638cd9947319734b9600ebad961.png[/img] [b]p5.[/b] Find all polynomials $f(x)$ such that $(x-2) f(x+1) - (x+1) f(x) = 0$ for all $x$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$\mathbb{Q}$ is set of all rational numbers. Find all functions $f:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}$ such that for all $x$, $y$, $z$ $\in\mathbb{Q}$ satisfy $f(x,y)+f(y,z)+f(z,x)=f(0,x+y+z)$

Find all continious $f: \mathbb R\longrightarrow\mathbb R$ that for any $x,y$ \[f(x)+f(y)+f(xy)=f(x+y+xy)\]

A subset $Q \subset H^s(\mathbb{R})$ is said to be equicontinuous if for any $\varepsilon>0$, $\exists \delta>0$ such that \[\|f(x+h)-f(x)\|_{H^s}<\varepsilon, \quad \forall \vert h\vert<\delta, \quad f \in Q.\] Fix $r<s$, given a bounded sequence of functions $f_n \in H^s(\mathbb{R}$. If $f_n$ converges in $H^r(\mathbb{R})$ and equicontinuous in $H^s(\mathbb{R})$, show that it also converges in $H^s(\mathbb{R})$.

Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$: \[ f(3mn \plus{} m \plus{} n) \equal{} 4f(m)f(n) \plus{} f(m) \plus{} f(n). \]

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]

Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y\in(1,\infty)$ and $x^2\le y\le x^3,$ then $(f(x))^2\le f(y) \le (f(x))^3.$

Let $f:\mathbb{R}\rightarrow[0,\infty )$ be a function with the property that $|f(x)-f(y)|\le |x-y|$ for every $x,y\in\mathbb{R}$. Show that: a) If $\lim_{n\rightarrow \infty} f(x+n)=\infty$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\infty$. b) If $\lim_{n\rightarrow \infty} f(x+n)=\alpha ,\alpha\in[0,\infty )$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\alpha$.

Let $f(x)$ be a function satisfying $f(x+1)-f(x)=2x+1 (x \in \mathbb{R})$.In addition, $|f(x)|\le 1$ holds for $x\in [0,1]$. Prove that $|f(x)|\le 2+x^2$ holds for $x \in \mathbb{R}$.