Let $ \mathbb{R}$ be the set of real numbers. Does there exist a function $ f: \mathbb{R} \mapsto \mathbb{R}$ which simultaneously satisfies the following three conditions? [b](a)[/b] There is a positive number $ M$ such that $ \forall x:$ $ \minus{} M \leq f(x) \leq M.$ [b](b)[/b] The value of $f(1)$ is $1$. [b](c)[/b] If $ x \neq 0,$ then \[ f \left(x \plus{} \frac {1}{x^2} \right) \equal{} f(x) \plus{} \left[ f \left(\frac {1}{x} \right) \right]^2 \]

Find all positive integers $m$ and $n$ such that the inequality: \[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \] is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.

Let $R$ be a square region and $n\ge4$ an integer. A point $X$ in the interior of $R$ is called $n\text{-}ray$ partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional? $\textbf{(A)}\,1500 \qquad\textbf{(B)}\,1560 \qquad\textbf{(C)}\,2320 \qquad\textbf{(D)}\,2480 \qquad\textbf{(E)}\,2500$

Let $p = 2^{16}+1$ be a prime, and let $S$ be the set of positive integers not divisible by $p$. Let $f: S \to \{0, 1, 2, ..., p-1\}$ be a function satisfying \[ f(x)f(y) \equiv f(xy)+f(xy^{p-2}) \pmod{p} \quad\text{and}\quad f(x+p) = f(x) \] for all $x,y \in S$. Let $N$ be the product of all possible nonzero values of $f(81)$. Find the remainder when when $N$ is divided by $p$. [i]Proposed by Yang Liu and Ryan Alweiss[/i]

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 $y$.

Let $S$ be the set of interior points of a sphere and $C$ be the set of interior points of a circle. Find, with proof, whether there exists a function $f:S\rightarrow C$ such that $d(A,B)\le d(f(A),f(B))$ for any two points $A,B\in S$ where $d(X,Y)$ denotes the distance between the points $X$ and $Y$. [i]Marius Cavachi[/i]

Determine all functions $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ that satisfies the equation $$f\left(\frac{x+y+z}{3},\frac{a+b+c}{3}\right)=f(x,a)f(y,b)f(z,c)$$ for any real numbers $x,y,z,a,b,c$ such that $az+bx+cy\neq ay+bz+cx$.

Let $f$ be a function with the following properties: 1) $f(n)$ is defined for every positive integer $n$; 2) $f(n)$ is an integer; 3) $f(2)=2$; 4) $f(mn)=f(m)f(n)$ for all $m$ and $n$; 5) $f(m)>f(n)$ whenever $m>n$. Prove that $f(n)=n$.

Let $f: \ [0,\ 1] \rightarrow \mathbb{R}$ be an increasing function satisfying the following conditions: a) $f(0)=0$; b) $f\left(\frac{x}{3}\right)=\frac{f(x)}{2}$; c) $f(1-x)=1-f(x)$. Determine $f\left(\frac{18}{1991}\right)$.

Morteza places a function $[0,1]\to [0,1]$ (that is a function with domain [0,1] and values from [0,1]) in each cell of an $n \times n$ board. Pavel wants to place a function $[0,1]\to [0,1]$ to the left of each row and below each column (i.e. to place $2n$ functions in total) so that the following condition holds for any cell in this board: If $h$ is the function in this cell, $f$ is the function below its column, and $g$ is the function to the left of its row, then $h(x) = f(g(x))$ for all $x \in [0, 1]$. Prove that Pavel can always fulfil his plan.

Let $n\ge 4$ be a positive integer and let $M$ be a set of $n$ points in the plane, where no three points are collinear and not all of the $n$ points being concyclic. Find all real functions $f:M\to\mathbb{R}$ such that for any circle $\mathcal{C}$ containing at least three points from $M$, the following equality holds: \[\sum_{P\in\mathcal{C}\cap M} f(P)=0\] [i]Dorel Mihet[/i]

Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have : $$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$ Who has the winning strategy. Proposed by [i]Alireza Haghi[/i]

Let $ R$ be a bounded domain of area $ t$ in the plane, and let $ C$ be its center of gravity. Denoting by $ T_{AB}$ the circle drawn with the diameter $ AB$, let $ K$ be a circle that contains each of the circles $ T_{AB} \;(A,B \in R)$. Is it true in general that $ K$ contains the circle of area $ 2t$ centered at $ C$? [i]J. Szucs[/i]

Let $a$ and $b$ be real numbers in the interval $\left(0,\tfrac{1}{2}\right)$, and let $g$ be a continuous real-valued function such that $g(g(x))=ag(x)+bx$ for all real $x$. Prove that $g(x)=cx$ for some constant $c$.

Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a differentiable function, with continuous derivative, and let \[ s_{n}=\sum_{k=1}^{n}f\left( \frac{k}{n}\right) \] Prove that the sequence $(s_{n+1}-s_{n})_{n\in{\mathbb{N}}^{\ast}}$ converges to $\int_{0}^{1}f(x)\mathrm{d}x$.

Find the strictly monotone functions $ f:\{ 0\}\cup\mathbb{N}\longrightarrow\{ 0\}\cup\mathbb{N} $ that satisfy the following two properties: $ \text{(i)} f(2n)=n+f(n), $ for any nonnegative integers $ n. $ $ \text{(ii)} f(n) $ is a perfect square if and only if $ n $ is a perfect square.

Let $c\ge 0$ be a real number. Give a complete description with proof of the set of all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that $f(x)=f(x^2+c)$ for all $x\in \mathbb{R}$.

suppose that $v(x)=\sum_{p\le x,p\in \mathbb P}log(p)$ (here $\mathbb P$ denotes the set of all positive prime numbers). prove that the two statements below are equivalent: [b]a)[/b] $v(x) \sim x$ when $x \longrightarrow \infty$ [b]b)[/b] $\pi (x) \sim \frac{x}{ln(x)}$ when $x \longrightarrow \infty$. (here $\pi (x)$ is number of the prime numbers less than or equal to $x$).

Consider the region $A$ in the complex plane that consists of all points $z$ such that both $\frac{z}{40}$ and $\frac{40}{\overline{z}}$ have real and imaginary parts between $0$ and $1$, inclusive. What is the integer that is nearest the area of $A$?

Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.

Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$ for all positive real numbers $x, y, z$. [i]Fajar Yuliawan, Indonesia[/i]

Let $ f: \mathbb{R}\rightarrow \mathbb{R}$ a strictly increasing function such that $ f\circ f$ is continuos. Prove that $ f$ is continuos.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$?

With about six hours left on the van ride home from vacation, Wendy looks for something to do. She starts working on a project for the math team. There are sixteen students, including Wendy, who are about to be sophomores on the math team. Elected as a math team officer, one of Wendy's jobs is to schedule groups of the sophomores to tutor geometry students after school on Tuesdays. The way things have been done in the past, the same number of sophomores tutor every week, but the same group of students never works together. Wendy notices that there are even numbers of groups she could select whether she chooses $4$ or $5$ students at a time to tutor geometry each week: \begin{align*}\dbinom{16}4&=1820,\\\dbinom{16}5&=4368.\end{align*} Playing around a bit more, Wendy realizes that unless she chooses all or none of the students on the math team to tutor each week that the number of possible combinations of the sophomore math teamers is always even. This gives her an idea for a problem for the $2008$ Jupiter Falls High School Math Meet team test: \[\text{How many of the 2009 numbers on Row 2008 of Pascal's Triangle are even?}\] Wendy works the solution out correctly. What is her answer?

Let $a$, $b$, $c$ be positive real numbers. Prove that \[ \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge\frac52\] and determine when equality holds.