Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$. [i]Sammy Luo and Alex Zhu.[/i]

Look at these fractions. At firs step we have $ \frac{0}{1}$ and $ \frac{1}{0}$, and at each step we write $ \frac{a\plus{}b}{c\plus{}d}$ between $ \frac{a}{b}$ and $ \frac{c}{d}$, and we do this forever \[ \begin{array}{ccccccccccccccccccccccccc}\frac{0}{1}&&&&&&&&\frac{1}{0}\\ \frac{0}{1}&&&&\frac{1}{1}&&&&\frac{1}{0}\\ \frac{0}{1}&&\frac{1}{2}&&\frac{1}{1}&&\frac{2}{1}&&\frac{1}{0}\\ \frac{0}{1}&\frac{1}{3}&\frac{1}{2}&\frac{2}{3}&\frac{1}{1}&\frac{3}{2}&\frac{2}{1}&\frac{3}{1}&\frac{1}{0}\\ &&&&\dots\end{array}\] a) Prove that each of these fractions is irreducible. b) In the plane we have put infinitely many circles of diameter 1, over each integer on the real line, one circle. The inductively we put circles that each circle is tangent to two adjacent circles and real line, and we do this forever. Prove that points of tangency of these circles are exactly all the numbers in part a(except $ \frac{1}{0}$). [img]http://i2.tinypic.com/4m8tmbq.png[/img] c) Prove that in these two parts all of positive rational numbers appear. If you don't understand the numbers, look at [url=http://upload.wikimedia.org/wikipedia/commons/2/21/Arabic_numerals-en.svg]here[/url].

Given a fixed positive integer $a\geq 9$. Prove: There exist finitely many positive integers $n$, satisfying: (1)$\tau (n)=a$ (2)$n|\phi (n)+\sigma (n)$ Note: For positive integer $n$, $\tau (n)$ is the number of positive divisors of $n$, $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with $n$, $\sigma (n)$ is the sum of positive divisors of $n$.

Let $P$ be the set of prime numbers. Consider a subset $M$ of $P$ with at least three elements. We assume that, for each non empty and finite subset $A$ of $M$, with $A \neq M$, the prime divisors of the integer $( \prod_{p \in A} ) - 1$ belong to $M$. Prove that $M = P$.

A word is a sequence of n letters of the alphabet {a, b, c, d}. A word is said to be complicated if it contains two consecutive groups of identic letters. The words caab, baba and cababdc, for example, are complicated words, while bacba and dcbdc are not. A word that is not complicated is a simple word. Prove that the numbers of simple words with n letters is greater than $2^n$, if n is a positive integer.

In Lineland there are $n\geq1$ towns, arranged along a road running from left to right. Each town has a [i]left bulldozer[/i] (put to the left of the town and facing left) and a [i]right bulldozer[/i] (put to the right of the town and facing right). The sizes of the $2n$ bulldozers are distinct. Every time when a left and right bulldozer confront each other, the larger bulldozer pushes the smaller one off the road. On the other hand, bulldozers are quite unprotected at their rears; so, if a bulldozer reaches the rear-end of another one, the first one pushes the second one off the road, regardless of their sizes. Let $A$ and $B$ be two towns, with $B$ to the right of $A$. We say that town $A$ can [i]sweep[/i] town $B$ [i]away[/i] if the right bulldozer of $A$ can move over to $B$ pushing off all bulldozers it meets. Similarly town $B$ can sweep town $A$ away if the left bulldozer of $B$ can move over to $A$ pushing off all bulldozers of all towns on its way. Prove that there is exactly one town that cannot be swept away by any other one.

All positive divisors of a positive integer $N$ are written on a blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In the firt move, the player $A$ erases $N$. If the last erased number is $d$, then the next player erases either a divisor of $d$ or a multiple of $d$. The player who cannot make a move loses. Determine all numbers $N$ for which $A$ can win independently of the moves of $B$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 2)[/i]

$a_1, a_2, ..., a_{95}$ are positive reals. Show that $\displaystyle \sum_{k=1}^{95}{a_k} \le 94+ \prod_{k=1}^{95}{\max{\{1,a_k\}}}$

Let Q+ and Z denote the set of positive rationals and the set of inte- gers, respectively. Find all functions f : Q+ → Z satisfying the following conditions: (i) f(1999) = 1; (ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+; (iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.

Define a function $f: \mathbb N \to \mathbb N$ by $f(1) = 1$, $f(n+1) = f(n) + 2^{f(n)}$ for every positive integer $n$. Prove that $f(1), f(2), \dots, f(3^{2013})$ leave distinct remainders when divided by $3^{2013}$.

$5$ points are given in the plane, any three non-collinear and any four non-concyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be [i]good[/i]. Let the number of good circles be $n$; find all possible values of $n$.

Let $ n$ be a nonzero positive integer. A set of persons is called a $ n$-balanced set if in any subset of $ 3$ persons there exists at least two which know each other and in each subset of $ n$ persons there are two which don't know each other. Prove that a $ n$-balanced set has at most $ (n \minus{} 1)(n \plus{} 2)/2$ persons.

Consider a regular $2^k$-gon with center $O$ and label its sides clockwise by $l_1,l_2,...,l_{2^k}$. Reflect $O$ with respect to $l_1$, then reflect the resulting point with respect to $l_2$ and do this process until the last side. Prove that the distance between the final point and $O$ is less than the perimeter of the $2^k$-gon. [i]Proposed by Hesam Rajabzade[/i]

Let $k,n\in\mathbb{N},0<k<n.$ Prove that \[\sum_{j=1}^k\binom{n}{j}=\binom{n}{1}+ \binom{n}{2}+\ldots + \binom{n}{k}\leq n^k.\]

Find all functions $f$ from positive integers to themselves such that: 1)$f(mn)=f(m)f(n)$ for all positive integers $m, n$ 2)$\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.

Assume that $k$ and $n$ are two positive integers. Prove that there exist positive integers $m_1 , \dots , m_k$ such that \[1+\frac{2^k-1}{n}=\left(1+\frac1{m_1}\right)\cdots \left(1+\frac1{m_k}\right).\] [i]Proposed by Japan[/i]

Let $S$ denote the set of rational numbers in the interval $(0,1)$. Determine, with proof, if there exists a subset $T$ of $S$ such that every element in $S$ can be uniquely written as the sum of finitely many distinct elements in $T$.

Let $ G$ be a finite non-commutative group of order $ t \equal{} 2^nm$, where $ n, m$ are positive and $ m$ is odd. Prove, that if the group contains an element of order $ 2^n$, then (i) $ G$ is not simple; (ii) $ G$ contains a normal subgroup of order $ m$.

Consider the function $ f: \mathbb{N}_0\to\mathbb{N}_0$, where $ \mathbb{N}_0$ is the set of all non-negative integers, defined by the following conditions : $ (i)$ $ f(0) \equal{} 0$; $ (ii)$ $ f(2n) \equal{} 2f(n)$ and $ (iii)$ $ f(2n \plus{} 1) \equal{} n \plus{} 2f(n)$ for all $ n\geq 0$. $ (a)$ Determine the three sets $ L \equal{} \{ n | f(n) < f(n \plus{} 1) \}$, $ E \equal{} \{n | f(n) \equal{} f(n \plus{} 1) \}$, and $ G \equal{} \{n | f(n) > f(n \plus{} 1) \}$. $ (b)$ For each $ k \geq 0$, find a formula for $ a_k \equal{} \max\{f(n) : 0 \leq n \leq 2^k\}$ in terms of $ k$.

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

Suppose $a_1, \dots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a set of integers with the following properties: (a) For $i=1, \dots, n$, $a_i \in S$. (b) For $i,j = 1, \dots, n$ (not necessarily distinct), $a_i - a_j \in S$. (c) For any integers $x,y \in S$, if $x+y \in S$, then $x-y \in S$. Prove that $S$ must be equal to the set of all integers.

There are $n$ rooms in a sauna, each has unlimited capacity. No room may be attended by a female and a male simultaneously. Moreover, males want to share a room only with males that they don't know and females want to share a room only with females that they know. Find the biggest number $k$ such that any $k$ couples can visit the sauna at the same time, given that two males know each other if and only if their wives know each other.

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$, \[f(a+d)+f(b-c)=f(a-d)+f(b+c).\] [i]Calvin Deng.[/i]

Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$

The Ababi alphabet consists of letters A and B, and the words in the Ababi language are precisely those that can be formed by the following two rules: 1) A is a word. 2) If s is a word, then $ s \oplus s$ and $ s \oplus \bar{s}$ are words, where $ \bar{s}$ denotes a word that is obtained by replacing all letters A in s with letters B, and vice versa; and $ x \oplus y$ denotes the concatenation of x and y. The Ululu alphabet consists also of letters A and B and the words in the Ululu language are precisely those that can be formed by the following two rules: 1) A is a word. 2) If s is a word, $ s \oplus s$ and $ s \oplus \bar{s}$ are words, where $ \bar{s}$ is defined as above and $ x \oplus y$ is a word obtained from words x and y of equal length by writing the letters of x and y alternatingly, starting from the first letter of x. Prove that the two languages consist of the same words.