Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant. [i]Victor Wang.[/i]

Let $k \in \mathbb{N}$, let $x_k$ denote the nearest integer to $\sqrt k$. Show that for each $m \in \mathbb {N}$, $$\sum_{k=1}^{m} \frac{1}{x_k} = f(m)+ \frac{m}{f(m)+1}$$, where $f(m)$ is the integer part of $\frac{\sqrt{4m-3}-1}{2}$

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for every positive integer $n$ the following is valid: If $d_1,d_2,\ldots,d_s$ are all the positive divisors of $n$, then $$f(d_1)f(d_2)\ldots f(d_s)=n.$$

Does there exist a function $f: \mathbb N \to \mathbb N$ such that for all integers $n \geq 2$, \[ f(f(n-1)) = f (n+1) - f(n)\, ?\]

In each of (a) to (d) you have to find a strictly increasing surjective function from A to B or prove that there doesn't exist any. (a) $A=\{x|x\in \mathbb{Q},x\leq \sqrt{2}\}$ and $B=\{x|x\in \mathbb{Q},x\leq \sqrt{3}\}$ (b) $A=\mathbb{Q}$ and $B=\mathbb{Q}\cup \{\pi \} $ In (c) and (d) we say $(x,y)>(z,t)$ where $x,y,z,t \in \mathbb{R}$ , whenever $x>z$ or $x=z$ and $y>t$. (c) $A=\mathbb{R}$ and $B=\mathbb{R}^2$ (d) $X=\{2^{-x}| x\in \mathbb{N}\}$ , then $A=X \times (X\cup \{0\})$ and $B=(X \cup \{ 0 \}) \times X$ (e) If $A,B \subset \mathbb{R}$ , such that there exists a surjective non-decreasing function from $A$ to $B$ and a surjective non-decreasing function from $B$ to $A$ , does there exist a surjective strictly increasing function from $B$ to $A$? Time allowed for this problem was 2 hours.

For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$. [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)[/i]

Find all real functions $f$ definited on positive integers and satisying: (a) $f(x+22)=f(x)$, (b) $f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)$ for all positive integers $x$ and $y$.

Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n))=n+1987$ for all $n$.

Find all real-valued functions $f$ on the reals such that $f(-x) = -f(x)$, $f(x+1) = f(x) + 1$ for all $x$, and $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$ for $x \not = 0$.

Let $\mathbb{R}$ be the set of all real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying for all $x,y\in\mathbb{R}$ the equation $f(x)+f(y)=f(f(x)f(y))$.

Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that: 1. $f$ has one-side limits in any $a\in \mathbb{R}$ and $f(a-0)\le f(a)\le f(a+0)$. 2. for any $a,b\in \mathbb{R},\ a<b$, we have $f(a-0)<f(b-0)$. Prove that $f$ is strictly increasing. [i]Mihai Piticari & Sorin Radulescu[/i]

Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$? [i]Proposed by Iceland.[/i]

Consider those functions $ f$ that satisfy $ f(x \plus{} 4) \plus{} f(x \minus{} 4) \equal{} f(x)$ for all real $ x$. Any such function is periodic, and there is a least common positive period $ p$ for all of them. Find $ p$. $ \textbf{(A)}\ 8\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 24\qquad \textbf{(E)}\ 32$

Let $M$ be an empty set of real numbers. For any $x \in M$ the functions $f: M\to M$ and $g: M\to M$ satisfy the relations $f (g (x)) = g (f (x)) = x$ and $f (x) + g (x) = x$. Show that $- x \in M$ ¸ and $f (-x) = -f (x)$ whatever $x \in M$.

Prove that $1\cdot2\cdot3\cdots 2002<\left(\frac{2003}{2}\right)^{2002}.$

Find all functions $ f:\mathbb{N} \rightarrow \mathbb{ N }_0 $ satisfy the following conditions: i) $ f(ab)=f(a)+f(b)-f(\gcd(a,b)), \forall a,b \in \mathbb{N} $ ii) For all primes $ p $ and natural numbers $ a $, $ f(a)\ge f(ap) \Rightarrow f(a)+f(p) \ge f(a)f(p)+1 $

Find all functions $g:R\rightarrow R$ for which there exists a strictly increasing function $ f:R\rightarrow R $ such that $f(x+y)=f(x)g(y)+f(y)$ $\forall x,y \in R$.

Given two positive integers $n$ and $m$ and a function $f : \mathbb{Z} \times \mathbb{Z} \to \left\{0,1\right\}$ with the property that \begin{align*} f\left(i, j\right) = f\left(i+n, j\right) = f\left(i, j+m\right) \qquad \text{for all } \left(i, j\right) \in \mathbb{Z} \times \mathbb{Z} . \end{align*} Let $\left[k\right] = \left\{1,2,\ldots,k\right\}$ for each positive integer $k$. Let $a$ be the number of all $\left(i, j\right) \in \left[n\right] \times \left[m\right]$ satisfying \begin{align*} f\left(i, j\right) = f\left(i+1, j\right) = f\left(i, j+1\right) . \end{align*} Let $b$ be the number of all $\left(i, j\right) \in \left[n\right] \times \left[m\right]$ satisfying \begin{align*} f\left(i, j\right) = f\left(i-1, j\right) = f\left(i, j-1\right) . \end{align*} Prove that $a = b$.

Let be a differentiable function $ f:\mathbb{R}_{> 0}\longrightarrow\mathbb{R}_{> 0} , $ and a primitive $ F:\mathbb{R}_{> 0}\longrightarrow\mathbb{R}_{> 0} $ of it such that $ F=f+f\cdot f. $ Show that: [b]a)[/b] $ f $ is nondecreasing. [b]b)[/b] $ \lim_{x\to\infty } f(x)/x =1/2 $ [i]Vasile Solovăstru[/i]

Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$. [i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]

Show that the improper integral \[ \lim_{B \rightarrow \infty} \displaystyle\int_{0}^{B} \sin (x) \sin (x^2) dx \] converges.

Find all positive integers $k$ for which there exists a nonlinear function $f:\mathbb{Z} \rightarrow\mathbb{Z}$ such that the equation $$f(a)+f(b)+f(c)=\frac{f(a-b)+f(b-c)+f(c-a)}{k}$$ holds for any integers $a,b,c$ satisfying $a+b+c=0$ (not necessarily distinct). [i]Evan Chen[/i]

Let $f(x)=\frac{\sin x}{x}$, for $x>0$, and let $n$ be a positive integer. Prove that $|f^{(n)}(x)|<\frac{1}{n+1}$, where $f^{(n)}$ denotes the $n^{\mathrm{th}}$ derivative of $f$. (Proposed by Alexander Bolbot, State University, Novosibirsk)

Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that: (i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$. (ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$. [i]Proposed by Yang Liu[/i]

Let $f(x)$ be a periodic function with periods $T$ and $1$($0<T<1$).Prove that: (1)If $T$ is rational,then there exists a prime $p$ such that $\frac{1}{p}$ is also a period of $f$; (2)If $T$ is irrational,then there exists a strictly decreasing infinite sequence {$a_n$},with $1>a_n>0$ for all positive integer $n$,such that all $a_n$ are periods of $f$.