Any of $6$ gossips has her own news. From time to time one of them makes a telephone call to some other gossip and they discuss fill the news they know. What the minimum number of the calls is necessary so as (for) all of them to know all the news?

Let $ABC$ be a triangle. Suppose that $X,Y$ are points in the plane such that $BX,CY$ are tangent to the circumcircle of $ABC$, $AB=BX,AC=CY$ and $X,Y,A$ are in the same side of $BC$. If $I$ be the incenter of $ABC$ prove that $\angle BAC+\angle XIY=180$.

Decide if there exists any number of $10$ digits such that rearranging $10,000$ times its digits results in $10,000$ different numbers that are multiples of $7$.

In quadrilateral $ABCD$, $m\angle B=m\angle C=120^\circ$, $AB=3$, $BC=4$, and $CD=5$. Find the area of $ABCD$. $\textbf{(A) }15\qquad\textbf{(B) }9\sqrt3\qquad\textbf{(C) }\dfrac{45\sqrt3}4\qquad\textbf{(D) }\dfrac{47\sqrt3}4\qquad\textbf{(E) }15\sqrt3$

If $ x \plus{} y \equal{} 1$, then the largest value of $ xy$ is: $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 0.5\qquad \textbf{(C)}\ \text{an irrational number about }{0.4}\qquad \textbf{(D)}\ 0.25\qquad \textbf{(E)}\ 0$

Given $a, b,c \ge 0, a + b + c = 1$, prove that $(a^2 + b^2 + c^2)^2 + 6abc \ge ab + bc + ac$ (I. Voronovich)

Ralph is standing along a road which heads straight east. If you go nine miles east, make a left turn, and travel seven miles north, you will find Pamela with her mountain bike. At exactly the same time that Ralph begins running eastward along the road at 6 miles per hour, Pamela begins biking in a straight line at 10 miles per hour. Pamela’s direction is chosen so that she will reach a point on the road where Ralph is running at exactly the same time Ralph reaches that same point. Let $M$ and $N$ be relatively prime positive integers such that $\frac{M}{N}$ is the number of hours that it takes Pamela and Ralph to meet. Find $M+N$.

Let $\alpha_{1}, \alpha_{2}, \cdots , \alpha_{n}$ be numbers in the interval $[0, 2\pi]$ such that the number $\displaystyle\sum_{i=1}^n (1 + \cos \alpha_{i})$ is an odd integer. Prove that \[\displaystyle\sum_{i=1}^n \sin \alpha_i \ge 1\]

An inscribed into a circle quadraliteral $ABCD$ is given. Points $M$ and $N$ lie on sides $AB$ and $CD$ such that $AK:KB=DM:MC$ and points $L$ and $N$ lie on sides $BC$ and $DA$ such that $BL:LC=AN:ND$. The circumcircle of the triangle $CML$ intersects diagonal $AC$ for the second time in point $P$. The circumcircle of triangle $DNM$ intersects diagonal $BD$ for the second time in point $Q$. Circumcircles of triangles $AKN$ and $BLK$ intersect for the second time in point $R$. Prove that the circumcircle of $PQR$ passes through the intersection of $AC$ and $BD$

Let $\Gamma_1$ and $\Gamma_2$ be concentric circles with radii $1$ and $2$, respectively. Four points are chosen on the circumference of $\Gamma_2$ independently and uniformly at random, and are then connected to form a convex quadrilateral. What is the probability that the perimeter of this quadrilateral intersects $\Gamma_1$? [i]Proposed by Yuan Yao.[/i]

The circles $C_1$ and $C_2$ with radii $r_1$ and $r_2$ touch the line $p$ at the point $P$. $C_1$ lies entirely within $C_2$. Line $q \perp p$ intersects $p$ at $S$ and touches $C_1$ at $R$. $q$ intersects $C_2$ at $M$ and $N$, where $N$ is between $R$ and $S$. a) Prove that line $PR$ bisects angle $\angle MPN$. b) Calculate the ratio $r_1 : r_2$ if line $PN$ bisects angle $\angle RPS$.

Show that for any $n$ we can find a set $X$ of $n$ distinct integers greater than 1, such that the average of the elements of any subset of $X$ is a square, cube or higher power.

Let be two natural numbers $ m,n\ge 2, $ two increasing finite sequences of real numbers $ \left( a_i \right)_{1\le i\le n} ,\left( b_j \right)_{1\le j\le m} , $ and the set $$ \left\{ a_i+b_j| 1\le i\le n,1\le j\le m \right\} . $$ Show that the set above has $ n+m-1 $ elements if and only if the two sequences above are arithmetic progressions and these have the same ratio.

Let $n$ and $k$ be positive integers with $k<n$. Let $P(x)$ be a polynomial of degree $n$ with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers $a_0,\,a_1,\,\ldots,\,a_k$ such that the polynomial $a_kx^k+\cdots+a_1x+a_0$ divides $P(x)$, the product $a_0a_1\cdots a_k$ is zero. Prove that $P(x)$ has a nonreal root.

Let $n$ be a positive integer. There are $2018n+1$ cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city $C$ and integer $1\le i\le 2018,$ there are exactly $n$ cities that are a distance $i$ away from $C.$ (The [i]distance[/i] between two cities is the least number of roads on any path between the two cities.) For which $n$ is it possible for Mark to achieve this? [i]Proposed by Michael Ren[/i]

Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a [i]maximal grid-aligned configuration[/i] on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find all possible values of $k(C)$ as a function of $n$. [i]Proposed by Holden Mui[/i]

Each student at a school divided 18 subjects into six disjoint triples. Could it happen that every triple of subjects is among the triples of exactly one student?

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

What is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors? [i]Ray Li[/i]

Let $AB$ be a line segment with length 2, and $S$ be the set of points $P$ on the plane such that there exists point $X$ on segment $AB$ with $AX = 2PX$. Find the area of $S$.

Find the smallest positive integer $n$ such that if we color in red $n$ arbitrary vertices of the cube , there will be a vertex of the cube which has the three vertices adjacent to it colored in red.

Let $n > 1$ be an integer. In a [i]configuration[/i] of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell [i]good[/i] if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations. [i]Proposed by Melek Güngör, Turkey[/i]

Let $k$ be a circle and $l$–line that is tangent to $k$ in point $P$. On $l$ from the two sides of $P$ are chosen arbitrary points $A$ and $B$. The tangents through $A$ and $B$ to $k$, different than $l$, intersect in point $C$. Find the geometric place of points $C$, when $A$ and $B$ change in such way so that $AP.BP$ is a constant.

The vertices of Durer's favorite regular decagon in clockwise order: $D_1, D_2, D_3, . . . , D_{10}$. What is the angle between the diagonals $D_1D_3$ and $D_2D_5$?