Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that (i): For different reals $x,y$, $f(x) \not= f(y)$. (ii): For all reals $x,y$, $f(x+f(f(-y)))=f(x)+f(f(y))$

Let $f(x)=|2\{x\} -1|$ where $\{x\}$ denotes the fractional part of $x$. The number $n$ is the smallest positive integer such that the equation $$nf(xf(x)) = x$$ has at least $2012$ real solutions $x$. What is $n$? $\textbf{Note:}$ the fractional part of $x$ is a real number $y= \{x\}$, such that $ 0 \le y < 1$ and $x-y$ is an integer. $ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 31\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}\ 62\qquad\textbf{(E)}\ 64 $

Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that the numbers \[n, f(n),f(f(n)),\dots,f^{m-1}(n)\] are distinct modulo $m$ for all integers $n,m$ with $m>1$. (Here $f^k$ is defined by $f^0(n)=n$ and $f^{k+1}(n)=f(f^{k}(n))$ for $k \ge 0$.)

If $f$ is a function from the set of positive integers to itself such that $f(x) \leq x^2$ for all natural $x$, and $f\left( f(f(x)) f(f(y))\right) = xy$ for all naturals $x$ and $y$. Find the number of possible values of $f(30)$. [i]Author: Alex Zhu[/i]

We're given the functions $f(x)=|x-1|-|x-2|$ and $g(x)=|x-3|$. a) Draw the graph of the function $f(x)$. b) Determine the area of the section enclosed by the functions $f(x)$ and $g(x)$.

A function $f$ satisfies the following condition for each $n\in N$: $f (1)+ f (2)+...+ f (n) = n^2 f (n)$. Find $f (1997)$ if $f (1) = 999$.

For a subset $ S$ of vertices of graph $ G$, let $ \Lambda(S)$ be the subset of all edges of $ G$ such that at least one of their ends is in $ S$. Suppose that $ G$ is a graph with $ m$ edges. Let $ d^*: V(G)\longrightarrow\mathbb N\cup\{0\}$ be a function such that a) $ \sum_{u}d^*(u)\equal{}m$. b) For each subset $ S$ of $ V(G)$: \[ \sum_{u\in S}d^*(u)\leq|\Lambda(S)|\] Prove that we can give directions to edges of $ G$ such that for each edge $ e$, $ d^\plus{}(e)\equal{}d^*(e)$.

Prove that if $ a,b,$ and $ c$ are positive real numbers, then \[ a^ab^bc^c \ge (abc)^{(a\plus{}b\plus{}c)/3}.\]

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\] $\textbf{(A) }278\qquad \textbf{(B) }279\qquad \textbf{(C) }280\qquad \textbf{(D) }281\qquad \textbf{(E) }282\qquad$

Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled: [b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label. [b]2.[/b] In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side. [b](a)[/b] Find the maximal $r$ for which such a labelling is possible. [b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of $r$, how many such labellings are there? [hide="Easier version (5th German TST 2006) - contains answer to the harder version"] [i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.[/hide] [i]Proposed by Federico Ardila, Colombia[/i]

Let $p(z)$ be a polynomial of degree $n,$ all of whose zeros have absolute value $1$ in the complex plane. Put $g(z)=\frac{p(z)}{z^{n/2}}.$ Show that all zeros of $g'(z)=0$ have absolute value $1.$

Find the continuous function satisfying the following equation. \[\int_0^x f(t)dt+\int_0^x tf(x-t)dt=e^{-x}-1.\] [i]1978 Shibaura Institute of Technology entrance exam[/i]

We have three functions. The first one is $y=\phi(x)$. The second one is the inverse function of the first one. The figure of the third funcion is symmetrical to the second one about line $x+y=0$. Then, the third function is $\text{(A)}y=-\phi(x)\qquad\text{(B)}y=-\phi(-x)\qquad\text{(C)}y=-\phi^{-1}(x)\qquad\text{(D)}y=-\phi^{-1}(x)$

Given is the function $f(x)=-x+\sqrt{(x+a)(x+b)}$, where $a$, $b$ are distinct given positive real numbers. Prove that for all real numbers $s\in (0,1)$ there exist only one positive real number $\alpha$ such that \[ f(\alpha)=\sqrt [s]{\frac{a^s+b^s}{2}} . \]

For nonzero real numbers $x$ and $y$, define $x\circ y = \tfrac{xy}{x+y}$. Compute \[2^1\circ \left(2^2\circ \left(2^3\circ\cdots\circ\left(2^{2016}\circ 2^{2017}\right)\right)\right).\]

If $\zeta$ denote the Riemann Zeta Function, and $s>1$ then $$\sum \limits_{k=1}^{\infty} \frac{1}{1+k^s}\geq \frac{\zeta (s)}{1+\zeta (s)}$$

Let $M(t)$ be measurable and locally bounded function, that is, \[M(t) \le C_{a,b}, \quad \forall 0 \le a \le t \le b<\infty\] with some constant $C_{a,b}$, from $[0,\infty)$ to $[0,\infty)$ such that \[M(t) \le 1+\int_0^t M(t-s)(1+t)^{-1}s^{-1/2} ds, \quad \forall t \ge 0.\] Show that \[M(t) \le 10+2\sqrt{5}, \quad \forall t \ge 0.\]

Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.

Let $f: \mathbb{Z}^2\to \mathbb{R}$ be a function. It is known that for any integer $C$ the four functions of $x$ \[f(x,C), f(C,x), f(x,x+C), f(x, C-x)\] are polynomials of degree at most $100$. Prove that $f$ is equal to a polynomial in two variables and find its maximal possible degree. [i]Remark: The degree of a bivariate polynomial $P(x,y)$ is defined as the maximal value of $i+j$ over all monomials $x^iy^j$ appearing in $P$ with a non-zero coefficient.[/i]

Let $a$, $b$, $n$ be positive integers with $a,b > 1$ and $\gcd(a,b) = 1$. Prove that $n$ divides $\phi\left(a^{n}+b^{n}\right)$.

Find the minimum value of \[|\sin{x} + \cos{x} + \tan{x} + \cot{x} + \sec{x} + \csc{x}|\] for real numbers $x$.

Does there exist a piecewise linear, continuous, surjective mapping $f: [0,1]\to [0,1]$ such that $f(0)=f(1)=0$, and for all positive integer $n$, $$2.0001^{(n-10)} <P_n(f)<2.9999^{(n+10)}$$holds, where $P_n(f)$ is the number of points $x$ such that $\underbrace{f(\dotsc f}_n(x)\dotsc )=x$?

For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]

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 $f(3)=g(3)$.