Suppose two functions $f(x)$ and $g(x)$ are defined for all $x$ with $2<x<4$ and satisfy: $2<f(x)<4,2<g(x)<4,f(g(x))=g(f(x))=x,f(x)\cdot g(x)=x^2$ for all $2<x<4$. Prove that $f(3)=g(3)$.

A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation $$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$ Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying: $ x^2 f(x)\plus{}f(1\minus{}x)\equal{}2x\minus{}x^4$ for all $ x \in \mathbb{R}$.

Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings: (1) $\sum_{v\in V} f(v)=|E|$; (2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$. Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.

Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition : 1) $f(-1)\geq f(1).$ 2) $x+f(x)$ is non decreasing function. 3) $\int_{-1}^ 1 f(x)\ dx=0.$ Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admits bounded primitives. Prove that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\left\{ \begin{matrix} x, & \quad x\le 0 \\ f(1/x)\cdot\ln x ,& \quad x>0 \end{matrix}\right. $$ admits primitives. [i]Florian Dumitrel[/i]

A $(2n + 1) \times (2n + 1)$ grid, with $n> 0$, is colored in such a way that each of the cell is white or black. A cell is called [i]special[/i] if there are at least $n$ other cells of the same color in its row, and at least another $n$ cells of the same color in its column. (a) Prove that there are at least $2n + 1$ special boxes. (b) Provide an example where there are at most $4n$ special cells. (c) Determine, as a function of $n$, the minimum possible number of special cells.

A real-valued function $f$ has the property that for all real numbers $a$ and $b,$ $$f(a + b) + f(a - b) = 2f(a) f(b).$$ Which one of the following cannot be the value of $f(1)?$ $ \textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } -1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } -2$

For a positive integer $n$, consider all nonincreasing functions $f : \{1,\hdots,n\}\to\{1,\hdots,n\}$. Some of them have a fixed point (i.e. a $c$ such that $f(c) = c$), some do not. Determine the difference between the sizes of the two sets of functions. [i]Remark.[/i] A function $f$ is [i]nonincreasing[/i] if $f(x) \geq f(y)$ holds for all $x \leq y$

Let $f(t)=t^3+t$. Decide if there exist rational numbers $x, y$ and positive integers $m, n$ such that $xy=3$ and: \begin{align*} \underbrace{f(f(\ldots f(f}_{m \ times}(x))\ldots)) = \underbrace{f(f(\ldots f(f}_{n \ times}(y))\ldots)). \end{align*}

Let $ f, g: \mathbb {R} \to \mathbb {R} $ function such that $ f (x + g (y)) = - x + y + 1 $ for each pair of real numbers $ x $ e $ y $. What is the value of $ g (x + f (y) $?

Let $n\geq 2$ and $\mathcal{M}$ be a subset of $S_n$ with at least two elements, and which is closed under composition. Consider a function $f:\mathcal{M}\rightarrow\mathbb{R}$ which satisfies $$|f(\sigma\tau)-f(\sigma)-f(\tau)|\leq 1,$$ for all $\sigma,\tau\in\mathcal{M}$. Prove that $$\max_{\sigma,\tau\in\mathcal{M}}|f(\sigma)-f(\tau)|\leq 2-\frac{2}{|\mathcal{M}|}.$$

Consider two positive real numbers $ a,b $ and the function $ f:(0,\infty )\longrightarrow\left( \sqrt{ab} ,\frac{a+b}{2} \right) $ defined as $ f(x)=-x+\sqrt{x^2+(a+b)x+ab}. $ Prove that it's bijective. [i]D.M. Bătineți-Giurgiu[/i] and [i]Neculai Stanciu[/i]

Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$

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

The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.

Consider the set $A = \{0, 1, 2, . . . , 2^{2n} - 1\}$. The function $f : A \rightarrow A$ is given by: $f(x_0 + 2x_1 + 2^2x_2 + ... + 2^{2n-1}x_{2n-1})=$$(1 - x_0) + 2x_1 + 2^2(1 - x_2) + 2^3x_3 + ... + 2^{2n-1}x_{2n-1}$ for every $0-1$ sequence $(x_0, x_1, . . . , x_{2n-1})$. Show that if $a_1, a_2, . . . , a_9$ are consecutive terms of an arithmetic progression, then the sequence $f(a_1), f(a_2), . . . , f(a_9)$ is not increasing.

Let $\mathbb{N}$ be the set of all natural numbers. Let $f:\mathbb{N} \to \mathbb{N}$ be a bijective function. Show that there exists three numbers $a$, $b$, $c$ in arithmatic progression such that $f(a)<f(b)<f(c)$

Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$, $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$ Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$.

Let be a natural number $ n, $ denote with $ C $ the square in the complex plane whose vertices are the affixes of $ 2n\pi\left( \pm 1\pm i \right) , $ and consider the set $$ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} $$ Prove the following implications. [b]a)[/b] $ \exists \alpha\in\mathbb{R}_{>0}\quad \forall z\in\partial C\quad \left| \cos z \right|\ge\alpha e^{|\text{Im}(z)|} $ [b]b)[/b] $ \forall f\in\Lambda\quad\frac{1}{2\pi i}\int_{\partial C} \frac{f(z)}{z^2\cos z} dz=f'(0)+\frac{4}{\pi^2}\sum_{p=-2n}^{2n-1} \frac{(-1)^{p+1} f(z-p)}{(1+2p)^2} $ [b]c)[/b] $ \forall f\in\Lambda\quad \sum_{p\in\mathbb{Z}}\frac{(-1)^pf\left( \frac{(1+2p)\pi}{2} \right)}{(1+2p)^2} =\frac{\pi^2 f'(0)}{4} $

Let $a_1\ge a_2\ge \ldots \ge a_n>0$ be $n$ reals. Prove the inequality \[a_1a_2\ldots a_{n-1}+(2a_2-a_1)(2a_3-a_2)\ldots (2a_n-a_{n-1})\ge 2a_2a_3\ldots a_n\]

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.

Answer the following questions: (1) Let $f(x)$ be a function such that $f''(x)$ is continuous and $f'(a)=f'(b)=0$ for some $a<b$. Prove that $f(b)-f(a)=\int_a^b \left(\frac{a+b}{2}-x\right)f''(x)dx$. (2) Consider the running a car on straight road. After a car which is at standstill at a traffic light started at time 0, it stopped again at the next traffic light apart a distance $L$ at time $T$. During the period, prove that there is an instant　for which the absolute value of the acceleration of the car is more than or equal to $\frac{4L}{T^2}.$

A student steps onto a stationary elevator and stands on a bathroom scale. The elevator then travels from the top of the building to the bottom. The student records the reading on the scale as a function of time. At what time(s) does the student have maximum downward velocity? $\textbf{(A)}$ At all times between $2 s$ and $4 s$ $\textbf{(B)}$ At $4 s$ only $\textbf{(C)}$ At all times between $4 s$ and $22 s$ $\textbf{(D)}$ At $22 s$ only $\textbf{(E)}$ At all times between $22 s$ and $24 s$