Find all functions $f\colon \mathbb{Q}\to \mathbb{R}_{\geq 0}$ such that for any two rational numbers $x$ and $y$ the following conditions hold [list] [*] $f(x+y)\leq f(x)+f(y)$, [*]$f(xy)=f(x)f(y)$, [*]$f(2)=1/2$. [/list]

Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that \[ f(xf(y)) \equal{} \frac {f(x)}{y} \] for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.

A row of $1000$ numbers is written on the blackboard. We write a new row, below the first according to the rule: We write under every number $a$ the natural number, indicating how many times the number $a$ is encountered in the first line. Then we write down the third line: under every number $b$ -- the natural number, indicating how many times the number $b$ is encountered in the second line, and so on. a) Prove that there is a line that coincides with the preceding one. b) Prove that the eleventh line coincides with the twelfth. c) Give an example of the initial line such, that the tenth row differs from the eleventh.

Define $f(x)$ to be the nearest integer to $x$, with the greater integer chosen if two integers are tied for being the nearest. For example, $f(2.3) = 2$, $f(2.5) = 3$, and $f(2.7) = 3$. Define $[A]$ to be the area of region $A$. Define region $R_n$, for each positive integer $n$, to be the region on the Cartesian plane which satisfies the inequality $f(|x|) + f(|y|) < n$. We pick an arbitrary point $O$ on the perimeter of $R_n$, and mark every two units around the perimeter with another point. Region $S_{nO}$ is defined by connecting these points in order. [b]a)[/b] Prove that the perimeter of $R_n$ is always congruent to $4 \pmod{8}$. [b]b)[/b] Prove that $[S_{nO}]$ is constant for any $O$. [b]c)[/b] Prove that $[R_n] + [S_{nO}] = (2n-1)^2$. [i]Proposed by Lewis Chen[/i]

Consider the equation \[ x^{2001}=y^x .\] [list] [*] Find all pairs $(x,y)$ of solutions where $x$ is a prime number and $y$ is a positive integer. [*] Find all pairs $(x,y)$ of solutions where $x$ and $y$ are positive integers.[/list] (Remember that $2001=3 \cdot 23 \cdot 29$.)

Given an $n\times n$ grid of dots, let $f(n)$ be the largest number of segments between adjacent dots which can be drawn such that (i) at most one segment is drawn between each pair of dots, and (ii) each dot has $1$ or $3$ segments coming from it. (For example, $f(4)=16$.) Compute $f(2000)$. [i]Proposed by David Stoner[/i]

Let $f(x)=\int_0^ x \frac{1}{1+t^2}dt.$ For $-1\leq x<1$, find $\cos \left\{2f\left(\sqrt{\frac{1+x}{1-x}}\right)\right\}.$ [i]2011 Ritsumeikan University entrance exam/Science and Technology[/i]

Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$

$ABC$ is a triangle. Find a point $X$ on $BC$ such that : area $ABX$ / area $ACX$ = perimeter $ABX$ / perimeter $ACX$.

On the plane are given a convex $n$-gon $P_1P_2....P_n$ and a point $Q$ inside it, not lying on any of its diagonals. Prove that if $n$ is even, then the number of triangles $P_iP_jP_k$ containing the point $Q$ is even.

Max has $2015$ jars labeled with the numbers $1$ to $2015$ and an unlimited supply of coins. Consider the following starting configurations: (a) All jars are empty. (b) Jar $1$ contains $1$ coin, jar $2$ contains $2$ coins, and so on, up to jar $2015$ which contains $2015$ coins. (c) Jar $1$ contains $2015$ coins, jar $2$ contains $2014$ coins, and so on, up to jar $2015$ which contains $1$ coin. Now Max selects in each step a number $n$ from $1$ to $2015$ and adds $n$ to each jar [i]except to the jar $n$[/i]. Determine for each starting configuration in (a), (b), (c), if Max can use a finite, strictly positive number of steps to obtain an equal number of coins in each jar. (Birgit Vera Schmidt)

Prove that for any positive integer $n$, \[\left\lfloor \frac{n}{3}\right\rfloor+\left\lfloor \frac{n+2}{6}\right\rfloor+\left\lfloor \frac{n+4}{6}\right\rfloor = \left\lfloor \frac{n}{2}\right\rfloor+\left\lfloor \frac{n+3}{6}\right\rfloor .\]

Let $i = \sqrt{-1}$. Define a sequence of complex numbers by $z_{1} = 0, z_{n+1} = z_{n}^{2}+i$ for $n \ge 1$. In the complex plane, how far from the origin is $z_{111}$? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt{2}\qquad\textbf{(C)}\ \sqrt{3}\qquad\textbf{(D)}\ \sqrt{110}\qquad\textbf{(E)}\ \sqrt{2^{55}} $

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Let $ ABCDEF$ be a regular hexagon. Let $ G$, $ H$, $ I$, $ J$, $ K$, and $ L$ be the midpoints of sides $ AB$, $ BC$, $ CD$, $ DE$, $ EF$, and $ AF$, respectively. The segments $ AH$, $ BI$, $ CJ$, $ DK$, $ EL$, and $ FG$ bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of $ ABCDEF$ be expressed as a fraction $ \frac {m}{n}$ where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

What is the largest number of Sundays can be in one year? Explain your answer.

Suppose $a_1, a_2, . . . , a_8$ are eight distinct integers from $\{1, 2, . . . , 16, 17\}$. Show that there is an integer $k > 0$ such that there are at least three different (not necessarily disjoint) pairs $(i, j)$ such that $a_i - a_j = k$. Also find a set of seven distinct integers from $\{1, 2, . . . , 16, 17\}$ such that there is no integer $k > 0$ with that property.

Let $C_1, ..., C_d$ be compact and connected sets in $R^d$, and suppose that each convex hull of $C_i$ contains the origin. Prove that for every i there is a $c_i \in C_i$ for which the origin is contained in the convex hull of the points $c_1, ..., c_d$.

Determine all primes $p$ such that $5^p + 4 p^4$ is a perfect square, i.e., the square of an integer.

Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.

Let $a_1, a_2, ....., a_{2003}$ be sequence of reals number. Call $a_k$ $leading$ element, if at least one of expression $a_k; a_k+a_{k+1}; a_k+a_{k+1}+a_{k+2}; ....; a_k+a{k+1}+a_{k+2}+....+a_{2003}$ is positive. Prove, that if exist at least one $leading$ element, then sum of all $leading$'s elements is positive. Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

Circle $\omega$ is inscribed in quadrilateral $ABCD$ such that $AB$ and $CD$ are not parallel and intersect at point $O.$ Circle $\omega_1$ touches the side $BC$ at $K$ and touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD;$ circle $\omega_2$ touches side $AD$ at $L$ and touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD.$ If $O,K,$ and $L$ are collinear$,$ then show that the midpoint of side $BC,AD,$ and the center of circle $\omega$ are also collinear.

Let $n$ be a positive integer. There are $\tfrac{n(n+1)}{2}$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each mark has the black side up. An [i]operation[/i] is to choose a line parallel to the sides of the triangle, and flipping all the marks on that line. A configuration is called [i]admissible [/i] if it can be obtained from the initial configuration by performing a finite number of operations. For each admissible configuration $C$, let $f(C)$ denote the smallest number of operations required to obtain $C$ from the initial configuration. Find the maximum value of $f(C)$, where $C$ varies over all admissible configurations.

Determine all pairs of positive integers $(a,b)$ such that \[ \dfrac{a^2}{2ab^2-b^3+1} \] is a positive integer.

Prove that for any real number $ x$ the following inequality is true: $ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$