On a blackboard there are $ n \geq 2, n \in \mathbb{Z}^{\plus{}}$ numbers. In each step we select two numbers from the blackboard and replace both of them by their sum. Determine all numbers $ n$ for which it is possible to yield $ n$ identical number after a finite number of steps.

Find all $n \in \mathbb{N}$ such that $3^{n}-n$ is divisible by $17$.

The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.

Find all prime numbers $ p,q,r$, such that $ \frac{p}{q}\minus{}\frac{4}{r\plus{}1}\equal{}1$

100 people from 25 countries, four from each countries, stay on a circle. Prove that one may partition them onto 4 groups in such way that neither no two countrymans, nor two neighbours will be in the same group.

In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. For what is the smallest amount of rubles you can rearrange chips in reverse order?

A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.

Let $ A$ be a subset of the set $ \{1, 2,\ldots,2006\}$, consisting of $ 1004$ elements. Prove that there exist $ 3$ distinct numbers $ a,b,c\in A$ such that $ gcd(a,b)$: a) divides $ c$ b) doesn't divide $ c$

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

On Christmas Eve, 1983, Dean Jixon, the famous seer who had made startling predictions of the events of the preceding year that the volcanic and seismic activities of $1980$ and $1981$ were connected with mathematics. The diminishing of this geological activity depended upon the existence of an elementary proof of the irreducibility of the polynomial \[P(x)=x^{1981}+x^{1980}+12x^{2}+24x+1983.\] Is there such a proof?

At the entrance to a cave is a rotating round table. On top of the table are $n$ identical barrels, evenly spaced along its circumference. Inside each barrel is a herring either with its head up or its head down. In a move, Ali Baba chooses from $1$ to $n$ of the barrels and turns them upside down. Then the table spins around. When it stops, it is impossible to tell which barrels have been turned over. The cave will open if the heads of the herrings in all $n$ barrels are up or are all down. Determine all values of $n$ for which Ali Baba can open the cave in a finite number of moves. [i](11 points)[/i]

A magician and his assistent are performing the following trick.There is a row of 12 empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of his choice, so that the assistent knows which boxes contain coins. The magician returns, and the assistant is allowed to open one box that does not contain a coin. Next, the magician selects 4 boxes, which are simultaneously opened. The goal of the magician is to open both boxes that contain coins. Devise a method that will allow the magician and his assistant to always succesfully perform the trick.

Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions \[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\] for all $x \in \mathbb{Q^+}.$

Suppose the sequence of nonnegative integers $a_1, a_2, \ldots, a_{1997}$ satisfies \[ a_i + a_j \leq a_{i+j} \leq a_i + a_j + 1 \] for all $i,j \geq 1$ with $i + j \leq 1997$. Show that there exists a real number $x$ such that $a_n = \lfloor nx \rfloor$ (the greatest integer $\leq nx$) for all $1 \leq n \leq 1997$.

Define real number $y$ as the fractional part of real number $x$ such that $0\leq y<1$ and $x-y$ is integer. Denote this by $<x>$. For real number $a$, define an infinite sequence $\{a_n\}\ (n=1,\ 2,\ 3,\ \cdots)$ inductively as follows. (i) $a_1=<a>$ (ii) If $a\n\neq 0$, then $a_{n+1}=\left<\frac{1}{a_n}\right>$, if $a_n=0$, then $a_{n+1}=0$. (1) For $a=\sqrt{2}$, find $a_n$. (2) For any natural number $n$, find real number $a\geq \frac 13$ such that $a_n=a$. (3) Let $a$ be a rational number. When we express $a=\frac{p}{q}$ with integer $p$, natural number $q$, prove that $a_n=0$ for any natural number $n\geq q$. [i]2011 Tokyo University entrance exam/Science, Problem 2[/i]

Let $n$ be a positive integer. On a blackboard is a circle, and around it $n$ non-negative integers are written. Raphinha plays a game in which an operation consists of erasing a number $a$ neighboring $b$ and $c$, with $b \geq c$, and writing in its place $b + c$ if $b + c \leq 5a/4$ and $b - c$ otherwise. Your goal is to make all the numbers on the board equal $0$. Find all $n$ such that Raphinha always manages to reach her goal.

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

Set $S_n = \sum_{p=1}^n (p^5+p^7)$. Determine the greatest common divisor of $S_n$ and $S_{3n}.$

Freddy writes down numbers $1, 2,\ldots ,n$ in some order. Then he makes a list of all pairs $(i, j)$ such that $1\le i<j\le n$ and the $i$-th number is bigger than the $j$-th number in his permutation. After that, Freddy repeats the following action while possible: choose a pair $(i, j)$ from the current list, interchange the $i$-th and the $j$-th number in the current permutation, and delete $(i, j)$ from the list. Prove that Freddy can choose pairs in such an order that, after the process finishes, the numbers in the permutation are in ascending order.

The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$. Note: In all the triangles the three vertices do not lie on a straight line.

In a simple graph $G$, we call $t$ pairwise adjacent vertices a $t$[i]-clique[/i]. If a vertex is connected with all other vertices in the graph, we call it a [i]central[/i] vertex. Given are two integers $n,k$ such that $\dfrac {3}{2} \leq \dfrac{1}{2} n < k < n$. Let $G$ be a graph on $n$ vertices such that [b](1)[/b] $G$ does not contain a $(k+1)$-[i]clique[/i]; [b](2)[/b] if we add an arbitrary edge to $G$, that creates a $(k+1)$-[i]clique[/i]. Find the least possible number of [i]central[/i] vertices in $G$.

Prove that there exists a triangle which can be cut into 2005 congruent triangles.

Each rational number is painted either white or red. Call such a coloring of the rationals [i]sanferminera[/i] if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions: [list=1][*]$xy=1$, [*]$x+y=0$, [*]$x+y=1$,[/list]we have $x$ and $y$ painted different colors. How many sanferminera colorings are there?

The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle. [asy] defaultpen(linewidth(0.7)); draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36)); draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61)); draw((34,36)--(34,45)--(25,45)); draw((36,36)--(36,38)--(34,38)); draw((36,38)--(41,38)); draw((34,45)--(41,45));[/asy]

Let $ A_0 \equal{} (a_1,\dots,a_n)$ be a finite sequence of real numbers. For each $ k\geq 0$, from the sequence $ A_k \equal{} (x_1,\dots,x_k)$ we construct a new sequence $ A_{k \plus{} 1}$ in the following way. 1. We choose a partition $ \{1,\dots,n\} \equal{} I\cup J$, where $ I$ and $ J$ are two disjoint sets, such that the expression \[ \left|\sum_{i\in I}x_i \minus{} \sum_{j\in J}x_j\right| \] attains the smallest value. (We allow $ I$ or $ J$ to be empty; in this case the corresponding sum is 0.) If there are several such partitions, one is chosen arbitrarily. 2. We set $ A_{k \plus{} 1} \equal{} (y_1,\dots,y_n)$ where $ y_i \equal{} x_i \plus{} 1$ if $ i\in I$, and $ y_i \equal{} x_i \minus{} 1$ if $ i\in J$. Prove that for some $ k$, the sequence $ A_k$ contains an element $ x$ such that $ |x|\geq\frac n2$. [i]Author: Omid Hatami, Iran[/i]