Let $N = 6^{n}$, where $n$ is a positive integer, and let $M = a^{N}+b^{N}$, where $a$ and $b$ are relatively prime integers greater than $1. M$ has at least two odd divisors greater than $1$ are $p,q$. Find the residue of $p^{N}+q^{N}\mod 6\cdot 12^{n}$.

Solve the equation system for real numbers: $$x_1+x_2=x_3^2$$ $$x_2+x_3=x_4^2$$ $$x_3+x_4=x_1^2$$ $$x_4+x_1=x_2^2$$

In the table $n \times n$ numbers from $1$ to $n$ are written in a spiral way. For which $n$ all the numbers on the main diagonal are distinct?

Equation $|x-2n|=k\sqrt{x}(n\in\mathbb{Z}_+)$ has two different real roots on $(2n-1,2n+1]$, then the range value of $k$ is $\text{(A)}k>0\qquad\text{(B)}0<k\leq\frac{1}{\sqrt{2n+1}}\qquad\text{(C)}\frac{1}{2n+1}<k\leq\frac{1}{\sqrt{2n+1}}$ $\text{(D)}$ none above

Find all non-constant polynoms $ f\in\mathbb{Q} [X] $ that don't have any real roots in the interval $ [0,1] $ and for which there exists a function $ \xi :[0,1]\longrightarrow\mathbb{Q} [X]\times\mathbb{Q} [X], \xi (x):=\left( g_x,h_x \right) $ such that $ h_x(x)\neq 0 $ and $ \int_0^x \frac{dt}{f(t)} =\frac{g_x(x)}{h_x(x)} , $ for all $ x\in [0,1] . $

Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$?

$ $ $ $ $ $ $ $There is a group of more than three airports. For any two airports $A, B$ belonging to this group, if there is an aircraft from $A$ to $ $ $B$, there is an aircraft from $B$ to $ $ $A$. For a list of different airports $A_0,A_1,...A_n$, define this list as a '[color=#00f]route[/color]' if there is an aircraft from $A_i$ to $A_{i+1}$ for each $i=0,1,...,n-1$. Also, define the beginning of this [color=#00f]route[/color] as $A_0$, the end as $A_n$, and the length as $n$. ($n\in \mathbb N$) $ $ $ $ $ $ $ $Now, let's say that for any three different pairs of airports $(A,B,C)$, there is always a [color=#00f]route[/color] $P$ that satisfies the following condition. $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ [b]Condition[/b]: $P$ begins with $A$ and ends with $B$, and does not include $C$. $ $ $ $ $ $When the length of the longest of the existing [color=#00f]route[/color]s is $M$ ($\ge 2$), prove that any two [color=#00f]route[/color]s of length $M$ contain at least two different airports simultaneously.

Let $\omega$ denote the circumscribed circle of triangle $ABC$, $I$ be its incenter, and $K$ be any point on arc $AC$ of $\omega$ not containing $B$. Point $P$ is symmetric to $I$ with respect to point $K$. Point $T$ on arc $AC$ of $\omega$ containing point $B$ is such that $\angle KCT = \angle PCI$. Show that the bisectors of angles $AKC$ and $ATC$ meet on line $CI$. [i](Proposed by Anton Trygub)[/i]

A variant triangle has fixed incircle and circumcircle. Prove that the radical center of its three excircles lies on a fixed circle and the circle's center is the midpoint of the line joining circumcenter and incenter. [i]proposed by Masoud Nourbakhsh[/i]

Let $ ABCD$ be a tetrahedron.Prove that if a point $ M$ in a space satisfies the relation: \begin{align*} MA^2 + MB^2 + CD^2 &= MB^2 + MC^2 + DA^2 \\ &= MC^2 + MD^2 + AB^2 \\ &= MD^2 + MA^2 + BC^2 . \end{align*} then it is found on the common perpendicular of the lines $ AC$ and $ BD$.

Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.

Let $a_1<a_2<...<a_t$ be $t$ given positive integers where no three form an arithmetic progression. For $k=t,t+1,...$ define $a_{k+1}$ to be the smallest positive integer larger than $a_k$ satisfying the condition that no three of $a_1,a_2,...,a_{k+1}$ form an arithmetic progression. For any $x\in\mathbb{R}^+$ define $A(x)$ to be the number of terms in $\{a_i\}_{i\ge 1}$ that are at most $x$. Show that there exist $c>1$ and $K>0$ such that $A(x)\ge c\sqrt{x}$ for any $x>K$.

Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$ a) Prove that the sequence consists only of natural numbers. b) Check if there are terms of the sequence divisible by $2011$.

For a natural number $x$ we define $f(x)$ to be the sum of all natural numbers less than $x$ and coprime with it. Let $m$ and $n$ be some natural numbers where $n$ is odd. Prove that there exist $x$, which is a multiple of $m$ and for which $f(x)$ is a perfect n-th power.

There are $n$ cards numbered and stacked in increasing order from up to down (i.e. the card in the top is the number 1, the second is the 2, and so on...). With this deck, the next steps are followed: -the first card (from the top) is put in the bottom of the deck. -the second card (from the top) is taken away of the deck. -the third card (from the top) is put in the bottom of the deck. -the fourth card (from the top) is taken away of the deck. - ... The proccess goes on always the same way: the card in the top is put at the end of the deck and the next is taken away of the deck, until just one card is left. Determine which is that card.

For an integer $n > 0$, denote by $\mathcal F(n)$ the set of integers $m > 0$ for which the polynomial $p(x) = x^2 + mx + n$ has an integer root. [list=a] [*] Let $S$ denote the set of integers $n > 0$ for which $\mathcal F(n)$ contains two consecutive integers. Show that $S$ is infinite but \[ \sum_{n \in S} \frac 1n \le 1. \] [*] Prove that there are infinitely many positive integers $n$ such that $\mathcal F(n)$ contains three consecutive integers. [/list] [i]Ivan Borsenco[/i]

Let $S=2+4+6+ \cdots +2N$, where $N$ is the smallest positive integer such that $S>1,000,000$. Then the sum of the digits of $N$ is: $\textbf{(A)}\ 27 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1$

Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1. [b]Note: Graph-Definition[/b]. A [b]graph[/b] consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x \equal{} v_{0},v_{1},v_{2},\cdots ,v_{m} \equal{} y\,$ such that each pair $ \,v_{i},v_{i \plus{} 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.

In the plane we have $16$ lines(not parallel and not concurrents), we have $120$ point(s) of intersections of this lines. Sebastian has to paint this $120$ points such that in each line all the painted points are with colour differents, find the minimum(quantity) of colour(s) that Sebastian needs to paint this points. If we have have $15$ lines(in this situation we have $105$ points), what's the minimum(quantity) of colour(s)?

An angle bisector $AD$ was drawn in triangle $ABC$. It turned out that the center of the inscribed circle of triangle $ABC$ coincides with the center of the inscribed circle of triangle $ABD$. Find the angles of the original triangle.

Let $ABC$ be a triangle such that $AB=9$, $BC=6$, and $AC=10$. $2$ points $D_1,D_2$ are labeled on $BC$ such that $BC$ is subdivided into $3$ equal segments; $4$ points $E_1,E_2,\dots,E_4$ are labeled on $AC$ such that $AC$ is subdivided into $5$ equal segments; and $8$ points $F_1,F_2,\dots,F_8$ are labeled on $AB$ such that $AB$ is subdivided into $9$ equal segments. All possible cevians are drawn from $A$ to each $D_i$; from $B$ to each $E_j$; and from $C$ to each $F_k$. At how many points in the interior of $\triangle ABC$ do at least $2$ cevians intersect?

Find all polynomials $P(x)$ with real coefficients that satisfy the relation $$1+P(x)=\frac{P(x-1)+P(x+1)}2.$$

On an exam with $80$ problems, Roger solved $68$ of them. What percentage of the problems did he solve?

Let $n\ge 2$ be an integer. Elwyn is given an $n\times n$ table filled with real numbers (each cell of the table contains exactly one number). We define a [i]rook set[/i] as a set of $n$ cells of the table situated in $n$ distinct rows as well as in n distinct columns. Assume that, for every rook set, the sum of $n$ numbers in the cells forming the set is nonnegative.\\ \\ By a move, Elwyn chooses a row, a column, and a real number $a,$ and then he adds $a$ to each number in the chosen row, and subtracts $a$ from each number in the chosen column (thus, the number at the intersection of the chosen row and column does not change). Prove that Elwyn can perform a sequence of moves so that all numbers in the table become nonnegative.

Let $a$ and $b$ be real numbers such that $ \frac {ab}{a^2 + b^2} = \frac {1}{4} $. Find all possible values of $ \frac {|a^2-b^2|}{a^2+b^2} $.