Let $k, m, n$ be integers such that $1<n\le m-1 \le k$. Determine the maximum size of a subset $S$ of the set $\{ 1,2, \cdots, k \}$ such that no $n$ distinct elements of $S$ add up to $m$.

Find all functions $f:\mathbb{Q}^{+} \to \mathbb{Q}^{+}$ such that for all $x\in \mathbb{Q}^+$: [list] [*] $f(x+1)=f(x)+1$, [*] $f(x^2)=f(x)^2$. [/list]

Graphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer $k$ such that no matter how these flights are arranged it is possible to travel between any two cities in Graphistan riding only GA flights as long as the absolute values of the difference between the number of flights originating and terminating at any city is not more than $k.$

Let $a_1,a_2,\ldots,a_n$ be a finite sequence of non negative integers, its subsequences are the sequences of the form $a_i,a_{i+1},\ldots,a_j$ with $1\le i\le j \le n$. Two subsequences are said to be equal if they have the same length and have the same terms, that is, two subsequences $a_i,a_{i+1},\ldots,a_j$ and $a_u,a_{u+1},\ldots a_v$ are equal iff $j-i=u-v$ and $a_{i+k}=a_{u+k}$ forall integers $k$ such that $0\le k\le j-1$. Finally, we say that a subsequence $a_i,a_{i+1},\ldots,a_j$ is palindromic if $a_{i+k}=a_{j-k}$ forall integers $k$ such that $0\le k \le j-i$ What is the greatest number of different palindromic subsequences that can a palindromic sequence of length $n$ contain?

Let $\left ( a_{n} \right )_{n \geq 1}$ be an increasing sequence of positive integers such that $a_1=1$, and for all positive integers $n$, $a_{n+1}\leq 2n$. Prove that for every positive $n$; there exists positive integers $p$ and $q$ such that $n=a_{p}-a_{q}$.

Sequence of real numbers $a_0,a_1,\dots,a_{1389}$ are called concave if for each $0<i<1389$, $a_i\geq\frac{a_{i-1}+a_{i+1}}2$. Find the largest $c$ such that for every concave sequence of non-negative real numbers: \[\sum_{i=0}^{1389}ia_i^2\geq c\sum_{i=0}^{1389}a_i^2\]

Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$. [i]Calvin Deng.[/i]

Determine the maximum number of three-term arithmetic progressions which can be chosen from a sequence of $n$ real numbers \[a_1<a_2<\cdots<a_n.\]

Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) \equal{} 1$, are elements of V. $ (a,b)$ is connected to $ (a,b \plus{} kab)$ by an edge and to $ (a \plus{} kab,b)$ by another edge for all integer k. Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.

Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$, \[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]

There are $ n\geq 5$ people in a party. Assume that among any three of them some two know each other. Show that one can select at least $ \frac{n}{2}$ people and arrange them at a round table so that each person sits between two of his/her acquaintances.

Let $N$ be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing $1$, and thereafter if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2n$ as long as he/she does not obtain a number greater than $N$. The player who writes $N$ wins. $(a)$ Determine which player has a winning strategy for $N=2005$. $(b)$ Determine which player has a winning strategy for $N=2004$. $(c)$ Find for how many integers $N\le 2005$ Barbara has a winning strategy.

Let $n\geq 2$ be an integer, and let $0 < a_1 < a_2 < \cdots < a_{2n+1}$ be real numbers. Prove the inequality \[ \sqrt[n]{a_1} - \sqrt[n]{a_2} + \sqrt[n]{a_3} - \cdots + \sqrt[n]{a_{2n+1}} < \sqrt[n]{a_1 - a_2 + a_3 - \cdots + a_{2n+1}}. \] [i]Bogdan Enescu, Romania[/i]

Let $n$ be a positive integer and let $T$ denote the collection of points $(x_1, x_2, \ldots, x_n) \in \mathbb R^n$ for which there exists a permutation $\sigma$ of $1, 2, \ldots , n$ such that $x_{\sigma(i)} - x_{\sigma(i+1) } \geq 1$ for each $i=1, 2, \ldots , n.$ Prove that there is a real number $d$ satisfying the following condition: For every $(a_1, a_2, \ldots, a_n) \in \mathbb R^n$ there exist points $(b_1, \ldots, b_n)$ and $(c_1,\ldots, c_n)$ in $T$ such that, for each $i = 1, . . . , n,$ \[a_i=\frac 12 (b_i+c_i) , \quad |a_i - b_i|  \leq d, \quad \text{and} \quad |a_i - c_i| \leq d.\]

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]