On regular hexagon $GOBEAR$ with side length $2$, bears are initially placed at $G, B, A$, forming an equilateral triangle. At time $t = 0$, all of them move clockwise along the sides of the hexagon at the same pace, stopping once they have each traveled $1$ unit. What is the total area swept out by the triangle formed by the three bears during their journey?

In a convex quadrilateral $ABCD$, $\angle ABD=30^\circ$, $\angle BCA=75^\circ$, $\angle ACD=25^\circ$ and $CD=CB$. Extend $CB$ to meet the circumcircle of triangle $DAC$ at $E$. Prove that $CE=BD$. [i]Proposed by BJ Venkatachala[/i]

A trapezoid $ABCD$ with bases $BC = a$ and $AD = 2a$ is drawn on the plane. Using only with a ruler, construct a triangle whose area is equal to the area of the trapezoid. With the help of a ruler you can draw straight lines through two known points. (Rozhkova Maria)

The diagonals $AC$ and $BD$ of a convex cyclic quadrilateral $ABCD$ intersect at point $E$. Given that $AB = 39, AE = 45, AD = 60$ and $BC = 56$, determine the length of $CD.$

Let $ABC$ be a triangle with $\angle CAB=60^{\circ}$ and with incenter $I$. Let points $D,E$ be on sides $AC,AB$, respectively, such that $BD$ and $CE$ are angle bisectors of angles $\angle ABC$ and $\angle BCA$, respectively. Show that $ID=IE$.

Altitudes $BE$ and $CF$ of triangle $ABC$ intersect in $H$. A perpendicular $HT$ from $H$ to $EF$ is drawn. Circumcircles $ABC$ and $BHT$ intersect at $B$ and $X$. Prove that $\angle TXA= \angle BAC$. [i]Vadzim Kamianetski[/i]

Vertices of a square $ABCD$ of side $\frac{25}4$ lie on a sphere. Parallel lines passing through points $A,B,C$ and $D$ intersect the sphere at points $A_1,B_1,C_1$ and $D_1$, respectively. Given that $AA_1=2$, $BB_1=10$, $CC_1=6$, determine the length of the segment $DD_1$.

Let $ \Omega$ is circle with radius $ R$ and center $ O$. Let $ \omega$ is a circle inside of the $ \Omega$ with center $ I$ radius $ r$. $ X$ is variable point of $ \omega$ and tangent line of $ \omega$ pass through $ X$ intersect the circle $ \Omega$ at points $ A,B$. A line pass through $ X$ perpendicular with $ AI$ intersect $ \omega$ at $ Y$ distinct with $ X$.Let point $ C$ is symmetric to the point $ I$ with respect to the line $ XY$.Find the locus of circumcenter of triangle $ ABC$ when $ X$ varies on $ \omega$

p1. Let $A = \{(x, y)|3x + 5y\ge 15, x + y^2\le 25, x\ge 0, x, y$ integer numbers $\}$. Find all pairs of $(x, zx)\in A$ provided that $z$ is non-zero integer. p2. A shop owner wants to be able to weigh various kinds of weight objects (in natural numbers) with only $4$ different weights. (For example, if he has weights $ 1$, $2$, $5$ and $10$. He can weighing $ 1$ kg, $2$ kg, $3$ kg $(1 + 2)$, $44$ kg $(5 - 1)$, $5$ kg, $6$ kg, $7$ kg, $ 8$ kg, $9$ kg $(10 - 1)$, $10$ kg, $11$ kg, $12$ kg, $13$ kg $(10 + 1 + 2)$, $14$ kg $(10 + 5 -1)$, $15$ kg, $16$ kg, $17$ kg and $18$ kg). If he wants to be able to weigh all the weight from $ 1$ kg to $40$ kg, determine the four weights that he must have. Explain that your answer is correct. p3. Given the following table. [img]https://cdn.artofproblemsolving.com/attachments/d/8/4622407a72656efe77ccaf02cf353ef1bcfa28.png[/img] Table $4\times 4$ ​​is a combination of four smaller table sections of size $2\times 2$. This table will be filled with four consecutive integers such that: $\bullet$ The horizontal sum of the numbers in each row is $10$ . $\bullet$ The vertical sum of the numbers in each column is $10$ $\bullet$ The sum of the four numbers in each part of $2\times 2$ which is delimited by the line thickness is also equal to $10$. Determine how many arrangements are possible. p4. A sequence of real numbers is defined as following: $U_n=ar^{n-1}$, if $n = 4m -3$ or $n = 4m - 2$ $U_n=- ar^{n-1}$, if $n = 4m - 1$ or $n = 4m$, where $a > 0$, $r > 0$, and $m$ is a positive integer. Prove that the sum of all the $ 1$st to $2009$th terms is $\frac{a(1+r-r^{2009}+r^{2010})}{1+r^2}$ 5. Cube $ABCD.EFGH$ is cut into four parts by two planes. The first plane is parallel to side $ABCD$ and passes through the midpoint of edge $BF$. The sceond plane passes through the midpoints $AB$, $AD$, $GH$, and $FG$. Determine the ratio of the volumes of the smallest part to the largest part.

In triangle $ABC$, touching points $A', B'$ of the incircle with $BC, AC$ and common point $G$ of segments $AA'$ and $BB'$ were marked. After this the triangle was erased. Restore it by the ruler and the compass.

Let $A_1A_2A_3A_4A_5$ be a convex 5-gon in which the coordinates of all of it's vertices are rational. For each $1\leq i \leq 5$ define $B_i$ the intersection of lines $A_{i+1}A_{i+2}$ and $A_{i+3}A_{i+4}$. ($A_i=A_{i+5}$) Prove that at most 3 lines from the lines $A_iB_i$ ($1\leq i \leq 5$) are concurrent. Time allowed for this problem was 75 minutes.

$P(x)$ is a polynomial of degree $3n$ such that \begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{eqnarray*} Determine $n$.

Let $ABCD$ be a trapezoid with bases $AB = 50$ and $CD = 125$, and legs $AD = 45$ and $BC = 60$. Find the area of the intersection between the circle centered at $B$ with radius $BD$ and the circle centered at $D$ with radius $BD$. Express your answer as a common fraction in simplest radical form and in terms of $\pi$.

A sphere is touched by all the four sides of a (space) quadrilateral. Prove that all the four touching points are in the same plane.

Let $ABC$ is triangle and $D \in AB$,$E \in AC$ such that $DE//BC$. Let $(ABC)$ meets with $(BDE)$ and $(CDE)$ at the second time $K,L$ respectively. $BK$ and $CL$ intersect at $T$. Prove that $TA$ is tangent to the $(ABC)$

İn triangle $ABC$ the bisector of $\angle BAC$ intersects the side $BC$ at the point $D$.The circle $\omega $ passes through $A$ and tangent to the side $BC$ at $D$.$AC$ and $\omega $ intersects at $M$ second time , $BM$ and $\omega $ intersects at $P$ second time. Prove that point $P$ lies on median of triangle $ABD$.

The lengths of the three sides of a $ \triangle ABC $ are rational. The altitude $ CD $ determines on the side $AB$ two segments $ AD $ and $ DB $. Prove that $ AD, DB $ are rational.

In triangle $ABC$, let $D$ lie on $AB$ such that $AD = AC$ and $\angle ADC = 20^{\circ}$. Let $l$ be a line through $B$ parallel to $CD$. Let $E$ lie on $l$ with $BE = AD$ so that $AE$ intersects segment $BC$ at $F$. If $\angle ABC = 10^{\circ}$, find the degree measure of $\angle FDC$.

In the convex hexagon $ABCDEF$, the angles at the vertices $B$, $C$, $E$, and $F$ are equal. Moreover, the equality \[AB+DE=AF+CD\] holds. Prove that the line $AD$ and the bisectors of the segments $BC$ and $EF$ have a common point.

Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.

Let $ABCD$ be a convex quadrilateral and let $K$, $L$, $M$, $N$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Let $NL$ and $KM$ meet at point $T$. Show that $8[DNTM] < [ABCD] < 8[DNTM]$, where $[P]$ denotes area of $P$.

Show that it is not possible to have a triangle with sides $a,b,$ and $c$ whose medians have length $\frac{2}{3}a, \frac{2}{3}b$ and $\frac{4}{5}c$.

Let $ABCD$ be a quadrilateral that can be inscribed in a circle whose diagonals are perpendicular. Denote by $P$ and $Q$ the feet of the perpendiculars through $D$ and $C$ respectively on the line $AB$, $X$ is the intersection point of the lines $AC$ and $DP$, $Y$ is the intersection point of the lines $BD$ and $CQ$. Show that $XY CD$ is a rhombus.

Given a quadrilateral $ABCD$ simultaneously inscribed and circumscribed, assume that none of its diagonals or sides is a diameter of the circumscribed circle. Let $P$ be the intersection point of the external bisectors of the angles near $A$ and $B$. Similarly, let $Q$ be the intersection point of the external bisectors of the angles $C$ and $D$. If $J$ and $O$ respectively are the incenter and circumcenter of $ABCD$ prove that $OJ\perp PQ$.

Let $\omega$ be the incircle of $\triangle ABC$ and $D,E,F$ be the tangency points on $BC ,CA, AB$. In $\triangle DEF$ let the altitudes from $E,F$ to $FD,DE$ intersect $AB, AC$ at $X ,Y$. Prove that the second intersection of $(AEX)$ and $(AFY)$ lies on $\omega$