Let the series $$s(n,x)=\sum \limits_{k= 0}^n \frac{(1-x)(1-2x)(1-3x)\cdots(1-nx)}{n!}$$ Find a real set on which this series is convergent, and then compute its sum. Find also $$\lim \limits_{(n,x)\to (\infty ,0)} s(n,x)$$

Determine all positive integers $n$ such that $\frac{n(n-1)}{2}-1$ divides $1^7+2^7+\dots +n^7$.

Point $M$ is the midpoint of the base $BC$ of trapezoid $ABCD$. On base $AD$, point $P$ is selected. Line $PM$ intersects line $DC$ at point $Q$, and the perpendicular from $P$ on the bases intersects line $BQ$ at point $K$. Prove that $\angle QBC = \angle KDA$.

Some of the towns in a country are connected with bidirectional paths, where each town can be reached by any other by going through these paths. From each town there are at least $n \geq 3$ paths. In the country there is no such route that includes all towns exactly once. Find the least possible number of towns in this country (Answer depends from $n$).

Prove that any group of people can be divided into two disjoint groups $A$ and $B$ such that any member from $A$ has at least half of his acquaintances in $B$ and any member from $B$ has at least half of his acquaintances in $A$ (acquaintance is reciprocal).

What is the greatest possible value of c such that $x^2+5x+c=0$ has at least one real solution?

(a) If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart? (b) What if "three'' is replaced by "nine''?

Let $ABC$ be an acute-angled triangle, $O$ its circumcenter and $H$ its orthocenter. The orthogonal projection of the vertex $A$ to the line $BC$ lies on the perpendicular bisector of the segment $AC$. Compute $\frac{CH}{BO}$ . (J. Willemson)

Consider the polynomial with real coefficients: \[ p(x) = a_{2008}x^{2008} + a_{2007}x^{2007} + \dots + a_1x + a_0 \] and it is given that its coefficients satisfy: \[ a_i + a_{i+1} = a_{i+2}, \quad i \in \{0,1,2,\dots,2006\} \] If $p(1) = 2008$ and $p(-1) = 0$, compute $a_{2008} - a_0$.

Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $4$ and passes through points $P$ and $R$. The area of the rectangle is [asy] pair A,B,C,D,P,Q,R; A = (0,4); B = (8,4); C = (8,0); D = (0,0); P = (2,2); Q = (4,2); R = (6,2); dot(A); dot(B); dot(C); dot(D); dot(P); dot(Q); dot(R); draw(A--B--C--D--cycle); draw(circle(P,2)); draw(circle(Q,2)); draw(circle(R,2)); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$P$",P,W); label("$Q$",Q,W); label("$R$",R,W); [/asy] $\text{(A)}\ 16 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 128$

Let $ABCD$ be a three-link broken line in space, all links of which are equal and $\angle BCD=90^o$. Find the distance from $A$ to the midpoint of $BD$, if $AD = a$.

$m(m>1)$ is an intenger, define $(a_n)$: $a_0=m,a_{n}=\varphi(a_{n-1})$ for all positive intenger $n$. If for all nonnegative intenger $k$, $a_{k+1}\mid a_k$, find all $m$ that is not larger than $2016$. Note: $\varphi(n)$ means Euler Function.

Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that $$f(a) \equiv g(a+m_p) \pmod p$$ holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that $$f(x)=g(x+r).$$

Evaluate \[2^{7}\times 3^{0}+2^{6}\times 3^{1}+2^{5}\times 3^{2}+\cdots+2^{0}\times 3^{7}.\] [i]Proposed by Nathan Xiong[/i]

In a tetrahedron, segments connecting the midpoints of heights with the orthocenters of the faces to which these heights are drawn intersect at one point. Prove that in such a tetrahedron all faces are equal or there are perpendicular edges. (Yu. Blinkov)

Let $\mathbb{R}_{>0}$ be the set of the positive real numbers. Find all functions $f:\mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ such that $$f(xy+f(x))=f(f(x)f(y))+x$$ for all positive real numbers $x$ and $y$.

Given any triangle $ABC.$ Let $O_1$ be it's circumcircle, $O_2$ be it's nine point circle, $O_3$ is a circle with orthocenter of $ABC$, $H$, and centroid $G$, be it's diameter. Prove that: $O_1,O_2,O_3$ share axis. (i.e. chose any two of them, their axis will be the same one, if $ABC$ is an obtuse triangle, the three circle share two points.)

How many solutions satisfying the condition $1 < x < 5$ does the equation $\{x[x]\} = 0.5$ have? (Here $[x]$ is the integer part of the number $x$, $\{x\} = x - [x]$ is the fractional part of the number $x$.)

Let $A=\{1,2,\ldots,n\}$. Find the number of unordered triples $(X,Y,Z)$ that satisfy $X\bigcup Y \bigcup Z=A$

Let $k{}$ be a positive integer. For every positive integer $n \leq 3^k$, denote $b_n$ the greatest power of $3$ that divides $C_{3^k}^n$. Compute $\sum_{n=1}^{3^k-1} \frac{1}{b_n}$.

Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.

Given a triangle $ABC$, let $D$ be the midpoint of the side $AC$ and let $M$ be the point that divides the segment $BD$ in the ratio $1/2$; that is, $MB/MD=1/2$. The rays $AM$ and $CM$ meet the sides $BC$ and $AB$ at points $E$ and $F$, respectively. Assume the two rays perpendicular: $AM\perp CM$. Show that the quadrangle $AFED$ is cyclic if and only if the median from $A$ in triangle $ABC$ meets the line $EF$ at a point situated on the circle $ABC$.

Determine all functions $f : \mathbb{R}^+\rightarrow\mathbb{R}$ that satisfy the following $f(1)=2008$, $|{f(x)}| \le x^2+1004^2$, $f\left (x+y+\frac{1}{x}+\frac{1}{y}\right )=f\left (x+\frac{1}{y}\right )+f\left (y+\frac{1}{x}\right ).$

An infinite grid with two rows is divided into unit squares. One of the cells in the second row is colored red and all other cells in the grid are white. Initially, we are in the red cell. In one move, we can move from one cell to an adjacent cell (sharing a side). Find the number of sequences of \( n \) moves such that no cell is visited more than once. (In particular, it is not allowed to return to the red cell after several moves.)

Let $m$ and $n$ be two positive integers, $m, n \ge 2$. Solve in the set of the positive integers the equation $x^n + y^n = 3^m$.