The integers $ c_{m,n}$ with $ m \geq 0, \geq 0$ are defined by \[ c_{m,0} \equal{} 1 \quad \forall m \geq 0, c_{0,n} \equal{} 1 \quad \forall n \geq 0,\] and \[ c_{m,n} \equal{} c_{m\minus{}1,n} \minus{} n \cdot c_{m\minus{}1,n\minus{}1} \quad \forall m > 0, n > 0.\] Prove that \[ c_{m,n} \equal{} c_{n,m} \quad \forall m > 0, n > 0.\]

For all real numbers $a$ and $b$, let \[a\Join b=\dfrac{a+b}{a-b}.\] Compute $1008\Join 1007$. [i] Proposed by David Altizio [/i]

Find all continous functions $f:[0,1]\to[0,2]$ with the property that $\left(\int_{\frac1{n+1}}^{\frac1n}xf(x)dx\right)^2=\int_{\frac1{n+1}}^{\frac1n}x^2f(x)dx, \forall n\in\mathbb N^*$. [i]Gabriel Marsanu, Andrei Nedelcu[/i]

Let $ M $ be a finite set of integers, and let be a function $ \varphi :\mathbb{Z}\longrightarrow\mathbb{Z} $ whose restriction to $ \mathbb{Z}\setminus M $ evaluates to a constant $ c, $ such that $$ 2\le |\varphi (M)|=|M|\neq \frac{1}{c}\cdot \sum_{\iota \in \varphi (M) } \iota . $$ Prove that $ \varphi $ is not a sum between an injective function and a surjective function. [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

$A=\{1,2,3,4,5,6,7,8\} .$ How many functions $f: A \rightarrow A$ are there such that $f(1)<f(2)<f(3)$ [list=1] [*] ${{8}\choose{3}}$ [*] ${{8}\choose{3}}5^{8}$ [*] ${{8}\choose{3}} 8^{5}$ [*] $\frac{8 !}{3 !} $ [/list]

Let f be a ternary operation on a set of at least four elements for which (1) $f ( x , x , y ) \equiv f ( x , y , x ) \equiv f( x , y , y ) \equiv x$ (2) $f ( x , y , z ) = f ( y , z , x ) = f ( y , x , z ) \in \{ x , y , z \}$ for pairwise distinct x,y,z. Prove that f is a nontrivial composition of g such that g is not a composition of f. (The n-variable operation g is trivial if $g(x_1, ..., x_n) \equiv x_i$ for some i ($1 \leq i \leq n$) )

Functions f and g are defined on the whole real line and are mutually inverse: g(f(x))=x, f(g(y))=y for all x, y. It is known that f can be written as a sum of periodic and linear functions: f(x)=kx+h(x) for some number k and a periodic function h(x). Show that g can also be written as a sum of periodic and linear functions. (A functions h(x) is called periodic if there exists a non-zero number d such that h(x+d)=h(x) for any x.)

Let $a$ be a positive integer. How many non-negative integer solutions x does the equation $\lfloor \frac{x}{a}\rfloor = \lfloor \frac{x}{a+1}\rfloor$ have? $\lfloor ~ \rfloor$ ---> [url=http://en.wikipedia.org/wiki/Floor_function]Floor Function[/url].

Find all real numbers $ x$ such that $ \lfloor x^3 \rfloor \equal{} 4x \plus{} 3$.

Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$. For $x\in \mathbb{R}$ we say that $f$ is [i]increasing in $x$[/i] if there exists $\epsilon_x > 0$ such that $f(x)\geq{f(a)}$, $\forall a\in (x-\epsilon_x,x)$ and $f(x)\leq f(b)$, $\forall b\in (x,x+\epsilon_x)$. $\textbf{(a)}$ Prove that if $f$ is increasing in $x$, $\forall x\in \mathbb{R}$ then $f$ is increasing over $\mathbb{R}$. $\textbf{(b)}$ We say that $f$ is [i]increasing to the left[/i] in $x$ if there exists $\epsilon_x > 0$ such that $f(x)\geq f(a) $, $ \forall a \in (x-\epsilon_x,x)$. Provide an example of a function $f: [0,1]\rightarrow \mathbb{R}$ for which there exists an infinite set $M \subset (0,1)$ such that $f$ is increasing to the left in every point of $M$, yet $f$ is increasing over no proper subinterval of $[0,1]$.

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

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which are differentiable in $0$ and satisfy the following inequality for all real numbers $x,y$ \[f(x+y)+f(xy)\geq f(x)+f(y).\][i]Dan Ștefan Marinescu and Mihai Piticari[/i]

What is the remainder when $2011^{(2011^{(2011^{(2011^{2011})})})}$ is divided by $19$ ? $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1$

Let $f$ be the function defined by $f(x) = 4x(1 - x)$. Let $n$ be a positive integer. Prove that there exist distinct real numbers $x_1$, $x_2$, $\ldots\,$, $x_n$ such that $x_{i + 1} = f(x_i)$ for each integer $i$ with $1 \le i \le n - 1$, and such that $x_1 = f(x_n)$.

If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds.

Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) [i]Proposed by Evan Chen[/i]

For $a>0$, find the minimum value of $I(a)=\int_1^e |\ln ax|\ dx.$

Solve the function $f: \Re \to \Re$: \[ f( x^{3} )+ f(y^{3}) = (x+y)(f(x^{2} )+f(y^{2} )-f(xy))\]

Let $g(t)$ and $h(t)$ be real, continuous functions for $t\geq 0.$ Show that any function $v(t)$ satisfying the differential inequality $$\frac{dv}{dt}+g(t)v \geq h(t),\;\; v(t)=c,$$ satisfies the further inequality $v(t)\geq u(t),$ where $$\frac{du}{dt}+g(t)u = h(t),\;\; u(t)=c.$$ From this, conclude that for sufficiently small $t>0,$ the solution of $$\frac{dv}{dt}+g(t)v = v^2 ,\;\; v(t)=c$$ may be written $$v=\max_{w(t)} \left( c e^{- \int_{0}^{t} |g(s)-2w(s)| \, ds} -\int_{0}^{t} e^{-\int_{0}^{t} |g(s')-2w(s')| \, ds'} w(s)^{2} ds \right),$$ where the maximum is over all continuous functions $w(t)$ defined over some $t$-interval $[0,t_0 ].$

A function $f:\mathbb R\to\mathbb R$ has the property that for every $x,y\in\mathbb R$ there exists a real number $t$ (depending on $x$ and $y$) such that $0<t<1$ and $$f(tx+(1-t)y)=tf(x)+(1-t)f(y).$$ Does it imply that $$f\left(\frac{x+y}2\right)=\frac{f(x)+f(y)}2$$ for every $x,y\in\mathbb R$?

Prove that: $\int_{-1}^1f^2(x)dx\ge \frac 1 2 (\int_{-1}^1f(x)dx)^2 +\frac 3 2(\int_{-1}^1xf(x)dx)^2$ Please give a proof without using even and odd functions. (the oficial proof uses those and seems to be un-natural) :D

Source: 1976 Euclid Part A Problem 10 ----- If $f$, $g$, $h$, and $k$ are functions and $a$ and $b$ are numbers such that $f(x)=(x-1)g(x)+3=(x+1)h(x)+1=(x^2-1)k(x)+ax+b$ for all $x$, then $(a,b)$ equals $\textbf{(A) } (-2,1) \qquad \textbf{(B) } (-1,2) \qquad \textbf{(C) } (1,1) \qquad \textbf{(D) } (1,2) \qquad \textbf{(E) } (2,1)$

Let $a$ be a real number in the open interval $(0,1)$. Let $n\geq 2$ be a positive integer and let $f_n\colon\mathbb{R}\to\mathbb{R}$ be defined by $f_n(x) = x+\frac{x^2}{n}$. Show that \[\frac{a(1-a)n^2+2a^2n+a^3}{(1-a)^2n^2+a(2-a)n+a^2}<(f_n \circ\ \cdots\ \circ f_n)(a)<\frac{an+a^2}{(1-a)n+a}\] where there are $n$ functions in the composition.

Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $|f(x)-f(y)| \leq |x-y|$, for all $x,y \in \mathbb{R}$. Prove that if for any real $x$, the sequence $x,f(x),f(f(x)),\ldots$ is an arithmetic progression, then there is $a \in \mathbb{R}$ such that $f(x)=x+a$, for all $x \in \mathbb R$.

Denote by $ M(r,f)$ the maximum modulus on the circle $ |z|\equal{}r$ of the transcendent entire function $ f(z)$, and by $ M_n(r,f)$ that of the $ nth$ partial sum of the power series of $ f(z)$. Prove that the existence of an entire function $ f_0(z)$ and a corresponding sequence of positive numbers $ r_1<r_2<...\rightarrow \plus{}\infty$ such that \[ \limsup_{n\rightarrow\infty} \frac{M_n(r_n,f_0)}{M(r_n,f_0)}\equal{}\plus{}\infty\] [P. Turan]