A group of $ 12 $ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $ k^\text{th} $ pirate to take a share takes $ \frac{k}{12} $ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $ 12^{\text{th}} $ pirate receive? $ \textbf{(A)} \ 720 \qquad \textbf{(B)} \ 1296 \qquad \textbf{(C)} \ 1728 \qquad \textbf{(D)} \ 1925 \qquad \textbf{(E)} \ 3850 $

Construct a quadrilateral with three sides $1$, $4$ and $3$ so that a circle could be circumscribed around it.

Three real numbers are given. Fractional part of the product of every two of them is $ 1\over 2$. Prove that these numbers are irrational. [i]Proposed by A. Golovanov[/i]

Medians $AD$, $BE$, and $CF$ of triangle $ABC$ meet at $G$ as shown. Six small triangles, each with vertex at $G$, are formed. We draw the circles inscribed in triangles $AFG$, $BDG$, and $CDG$ as shown. Prove that if these three circles are all congruent, then $ABC$ is equilateral. [asy] size(200); defaultpen(fontsize(10)); pair C=origin, B=(12,0), A=(3,14), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C); draw(A--B--C--A--D^^B--E^^C--F); draw(incircle(C,G,D)^^incircle(G,D,B)^^incircle(A,F,G)); pair point=G; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$G$", G, dir(7));[/asy]

Given the parallelogram $ABCD$. The circle $S_1$ passes through the vertex $C$ and touches the sides $BA$ and $AD$ at points $P_1$ and $Q_1$, respectively. The circle $S_2$ passes through the vertex $B$ and touches the side $DC$ at points $P_2$ and $Q_2$, respectively. Let $d_1$ and $d_2$ be the distances from $C$ and $B$ to the lines $P_1Q_1$ and $P_2Q_2$, respectively. Find all possible values of the ratio $d_1:d_2$. [i](I. Voronovich)[/i]

Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.

Consider a right triangle $ABC$, where $A$ is the right angle, and $AC > AB$. Points $E$ on $AC$ and $D$ on $BC$ are chosen so that$ AB = AE = BD$. Prove that the triangle $ADE$ is right if and only if the ratio $AB : AC : BC$ of sides of the triangle $ABC$ is $3 : 4 : 5$. (A. Parovan)

For a positive integer $n$ define $S_n=1!+2!+\ldots +n!$. Prove that there exists an integer $n$ such that $S_n$ has a prime divisor greater than $10^{2012}$.

Consider a function $f:\mathbb{Z}\to \mathbb{Z}$ such that: \[f(m^2+f(n))=f^2(m)+n,\ \forall m,n\in \mathbb{Z}\] Prove that: a)$f(0)=0$; b)$f(1)=1$; c)$f(n)=n,\ \forall n\in \mathbb{Z}$ [i]Lucian Dragomir[/i]

Suppose $a,b,c$ are real numbers so that $a+b+c=15$ and $ab+ac+bc=27$. Find the range of values that may be obtained by the expression $abc$.

Find all number triples $(x,y,z)$ such that when any of these numbers is added to the product of the other two, the result is 2.

Let $ABC$ be an acute triangle. Suppose that circle $\Gamma_1$ has it's center on the side $AC$ and is tangent to the sides $AB$ and $BC$, and circle $\Gamma_2$ has it's center on the side $AB$ and is tangent to the sides $AC$ and $BC$. The circles $\Gamma_1$ and $ \Gamma_2$ intersect at two points $P$ and $Q$. Show that if $A, P, Q$ are collinear, then $AB = AC$.

Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \[N=1111\cdots 11.\] Find the thousandth digit after the decimal point of $\sqrt N$.

Find a four digit number $M$ such that the number $N=4\times M$ has the following properties. (a) $N$ is also a four digit number (b) $N$ has the same digits as in $M$ but in reverse order.

Prove that for equation $$x^{2015} + y^{2015} = z^{2016}$$ there are infinitely many solutions where $x,y$ and $z$ are different natural numbers.

Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.

Let $p(x) = x^{2} + bx + c$, where $b$ and $c$ are integers. If $p(x)$ is a factor of both \[x^{4} + 6x^{2} + 25\quad\text{and}\quad 3x^{4} + 4x^{2} + 28x + 5,\] what is $p(1)$? $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 8 $

Find all prime numbers $p,q$ such that: $$p^4+p^3+p^2+p=q^2+q$$

Let $a > b > c$ be sides of a triangle and $h_a,h_b,h_c$ be the corresponding altitudes. Prove that $a+h_a > b+h_b > c+h_c$.

Gleb picked positive integers $N$ and $a$ ($a < N$). He wrote the number $a$ on a blackboard. Then each turn he did the following: he took the last number on the blackboard, divided the number $N$ by this last number with remainder and wrote the remainder onto the board. When he wrote the number $0$ onto the board, he stopped. Could he pick $N$ and $a$ such that the sum of the numbers on the blackboard would become greater than $100N$ ? Ivan Mitrofanov

Let $ \alpha$ be a projective plane and $ c$ a closed polygon on $ \alpha$. Prove that $ \alpha$ will be decomposed into two regions by $ c$ if and only if there exists a straight line $ g$ in $ \alpha$ which has an even number of points in common with $ c$.

The acronym AMC is shown in the rectangular grid below with grid lines spaced $1$ unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC$?$ [asy] import olympiad; unitsize(25); for (int i = 0; i < 3; ++i) { for (int j = 0; j < 9; ++j) { pair A = (j,i); } } for (int i = 0; i < 3; ++i) { for (int j = 0; j < 9; ++j) { if (j != 8) { draw((j,i)--(j+1,i), gray(0.6)+dashed); } if (i != 2) { draw((j,i)--(j,i+1), gray(0.6)+dashed); } } } draw((0,0)--(2,2),linewidth(2)); draw((2,0)--(2,2),linewidth(2)); draw((1,1)--(2,1),linewidth(2)); draw((3,0)--(3,2),linewidth(2)); draw((5,0)--(5,2),linewidth(2)); draw((4,1)--(3,2),linewidth(2)); draw((4,1)--(5,2),linewidth(2)); draw((6,0)--(8,0),linewidth(2)); draw((6,2)--(8,2),linewidth(2)); draw((6,0)--(6,2),linewidth(2)); [/asy] $\textbf{(A) } 17 \qquad \textbf{(B) } 15 + 2\sqrt{2} \qquad \textbf{(C) } 13 + 4\sqrt{2} \qquad \textbf{(D) } 11 + 6\sqrt{2} \qquad \textbf{(E) } 21$

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

In a competition, there were $2n+1$ teams. Every team plays exatly once against every other team. Every match finishes with the victory of one of the teams. We call cyclical a 3-subset of team ${ A,B,C }$ if $A$ won against $B$, $B$ won against $C$ , $C$ won against $A$. (a) Find the minimum of cyclical 3-subset (depending on $n$); (b) Find the maximum of cyclical 3-subset (depending on $n$).

Show that the following inequality hold for all $ k \geq 1$, real numbers $ a_1,a_2,...,a_k$, and positive numbers $ x_1,x_2,...,x_k.$ \[ \ln \frac {\sum\limits_{i \equal{} 1}^kx_i}{\sum\limits_{i \equal{} 1}^kx_i^{1 \minus{} a_i}} \leq \frac {\sum\limits_{i \equal{} 1}^ka_ix_i \ln x_i}{\sum\limits_{i \equal{} 1}^kx_i} . \] [i]L. Losonczi[/i]