Find all functions $ f: \mathbb{Q}^{\plus{}} \mapsto \mathbb{Q}^{\plus{}}$ such that: \[ f(x) \plus{} f(y) \plus{} 2xy f(xy) \equal{} \frac {f(xy)}{f(x\plus{}y)}.\]

Put $\mathbb{A}=\{ \mathrm{yes}, \mathrm{no} \}$. A function $f\colon \mathbb{A}^n\rightarrow \mathbb{A}$ is called a [i]decision function[/i] if (a) the value of the function changes if we change all of its arguments; and (b) the values does not change if we replace any of the arguments by the function value. A function $d\colon \mathbb{A}^n \rightarrow \mathbb{A}$ is called a [i]dictatoric function[/i], if there is an index $i$ such that the value of the function equals its $i$th argument. The [i]democratic function[/i] is the function $m\colon \mathbb{A}^3 \rightarrow \mathbb{A}$ that outputs the majority of its arguments. Prove that any decision function is a composition of dictatoric and democratic functions.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^3+f(y)\right)=x^2f(x)+y,$$for all $x,y\in\mathbb{R}.$ (Here $\mathbb{R}$ denotes the set of all real numbers.)

Given the positive real numbers $a_{1}<a_{2}<\cdots<a_{n}$, consider the function \[f(x)=\frac{a_{1}}{x+a_{1}}+\frac{a_{2}}{x+a_{2}}+\cdots+\frac{a_{n}}{x+a_{n}}\] Determine the sum of the lengths of the disjoint intervals formed by all the values of $x$ such that $f(x)>1$.

Prove the inequality \[ \sqrt {a^{1 \minus{} a}b^{1 \minus{} b}c^{1 \minus{} c}} \le \frac {1}{3} \] holds for all positive real numbers $ a$, $ b$ and $ c$ with $ a \plus{} b \plus{} c \equal{} 1$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f\left( {yf(x + y) + f(x)} \right) = 4x + 2yf(x + y)\] for all $x,y\in\mathbb{R}$. [i]Netherlands (Birgit van Dalen)[/i]

The game of $pebbles$ is played on an infinite board of lattice points $(i,j)$. Initially there is a $pebble$ at $(0,0)$. A move consists of removing a $pebble$ from point $(i,j)$and placing a $pebble$ at each of the points $(i+1,j)$ and $(i,j+1)$ provided both are vacant. Show taht at any stage of the game there is a $pebble$ at some lattice point $(a,b)$ with $0 \leq a+b \leq 3$

The function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties: \begin{eqnarray*} f(x,x) &=& x, \\ f(x,y) &=& f(y,x), \quad \text{and} \\ (x + y) f(x,y) &=& yf(x,x + y). \end{eqnarray*} Calculate $f(14,52)$.

Consider the expansion \[(1 + x + x^2 + x^3 + x^4)^{496} = a_0 + a_1x + \cdots + a_{1984}x^{1984}.\] [b](a)[/b] Determine the greatest common divisor of the coefficients $a_3, a_8, a_{13}, \ldots , a_{1983}.$ [b](b)[/b] Prove that $10^{340 }< a_{992} < 10^{347}.$

Let $a > 1$ and $r > 1$ be real numbers. (a) Prove that if $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ is a function satisfying the conditions (i) $f (x)^{2}\leq ax^{r}f (\frac{x}{a})$ for all $x > 0$, (ii) $f (x) < 2^{2000}$ for all $x < \frac{1}{2^{2000}}$, then $f (x) \leq x^{r}a^{1-r}$ for all $x > 0$. (b) Construct a function $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ satisfying condition (i) such that for all $x > 0, f (x) > x^{r}a^{1-r}$ .

Find all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ such that $$f(x+f(y+1))+f(xy)=f(x+1)(f(y)+1)$$ for any $x,y$ integers.

The function $f$ is given by the table \[\begin{array}{|c||c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 4 & 1 & 3 & 5 & 2 \\ \hline \end{array}\] If $u_0=4$ and $u_{n+1}=f(u_n)$ for $n\geq 0$, find $u_{2002}$. $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

For natural number $n$, define the function $f(n)$ to be the number you get by $f(n)$ adding the digits of the number $n$. For example, $f(16)=7$, $f(f(78))=6$, and $f(f(f(5978)))=2$. Find the least natural number $n$ such that $f(f(f(n)))$ is not a one-digit number.

Prove that the function $f : \mathbb{N}\longrightarrow \mathbb{Z}$ defined by $f(n) = n^{2007}-n!$, is injective.

Let $ ABC$ be an acute-angled triangle. Let $ L$ be any line in the plane of the triangle $ ABC$. Denote by $ u$, $ v$, $ w$ the lengths of the perpendiculars to $ L$ from $ A$, $ B$, $ C$ respectively. Prove the inequality $ u^2\cdot\tan A \plus{} v^2\cdot\tan B \plus{} w^2\cdot\tan C\geq 2\cdot S$, where $ S$ is the area of the triangle $ ABC$. Determine the lines $ L$ for which equality holds.

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

The minimal value of $ f(x) \equal{} \sqrt{a^2 \plus{} x^2} \plus{} \sqrt{(x\minus{}b)^2 \plus{} c^2}$ is A. $ a\plus{}b\plus{}c$ B. $ \sqrt{a^2 \plus{} (b \plus{} c)^2}$ C. $ \sqrt{b^2 \plus{} (a\plus{}c)^2}$ D. $ \sqrt{(a\plus{}b)^2 \plus{} c^2}$ E. None of these

We say that a function $f : N \to N$ is increasing if $f(n) < f(m)$ whenever $n < m$, multiplicative if $f(nm) = f(n)f(m)$ whenever $n$ and $m$ are coprime, and completely multiplicative if $f(nm) = f(n)f(m)$ for all $n,m$. (a) Prove that if $f$ is increasing then $f(n) \ge n$ for each $n$. (b) Prove that if $f$ is increasing and completely multiplicative and $f(2) = 2$, then $f(n) = n$ for all $n$. (c) Does (b) remain true if the word ”completely” is omitted?

Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.

An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions $f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$). A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.

Given $n \in\mathbb{N}$, find all continuous functions $f : \mathbb{R}\to \mathbb{R}$ such that for all $x\in\mathbb{R},$ \[\sum_{k=0}^{n}\binom{n}{k}f(x^{2^{k}})=0. \]

Given $1000$ linear functions $f_k(x)=p_k x + q_k$ where $k = 1 , 2 ,... , 1000$, it is necessary to evaluate their composite $f(x) =f_1 (f_2(f_3 ... f_{1000}(x)...))$ at the point $x_0$ . Prove that this can be done in no more than $30$ steps, where at each step one may execute simultaneously any number of arithmetic operations on pairs of numbers obtained from the previous step (at the first step one may use the numbers $p_1 , p_2 ,... ,p_{1000}, q_l , q_2 ,... ,q_{1000} , x_o$). {S. Fomin, Leningrad)

$m(m>1)$ is an intenger, define $(a_n)$: $a_0=m,a_{n}=\varphi(a_{n-1})$ for all positive intenger $n$. If for all nonnegative intenger $k$, $a_{k+1}\mid a_k$, find all $m$ that is not larger than $2016$. Note: $\varphi(n)$ means Euler Function.

Determine all functions $f : \mathbb{R}^+\rightarrow\mathbb{R}$ that satisfy the following $f(1)=2008$, $|{f(x)}| \le x^2+1004^2$, $f\left (x+y+\frac{1}{x}+\frac{1}{y}\right )=f\left (x+\frac{1}{y}\right )+f\left (y+\frac{1}{x}\right ).$