An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding $2m$ to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount $m$ and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.

In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.

Let $AA_1, BB_1, CC_1$ be the heights of triangle $ABC$ and $H$ be its orthocenter. Liune $\ell$ parallel to $AC$, intersects straight lines $AA_1$ and $CC_1$ at points $A_2$ and $C_2$, respectively. Suppose that point $B_1$ lies outside the circumscribed circle of triangle $A_2 HC_2$. Let $B_1P$ and $B_1T$ be tangent to of this circle. Prove that points $A_1, C_1, P$, and $T$ are cyclic.

Let be a point $ P $ in the interior of a parallelogram $ ABCD $ such that $ \angle PAD=\angle PCD. $ Prove that the bisectors of $ \angle BAD $ and $ \angle BPD $ are parallel.

Let $S$ be a shape in the plane which is obtained as a union of finitely many unit squares. Prove that the ratio of the perimeter and the area of $S$ is at most $8$.

$AB$ is a chord of a circle with center $O$, $M$ is the midpoint of $AB$. A non-diameter chord is drawn through $M$ and intersects the circle at $C$ and $D$. The tangents of the circle from points $C$ and $D$ intersect line $AB$ at $P$ and $Q$, respectively. Prove that $PA$ = $QB$.

Let $\gamma$ be a circle and $l$ a line in its plane. Let $K$ be a point on $l$, located outside of $\gamma$. Let $KA$ and $KB$ be the tangents from $K$ to $\gamma$, where $A$ and $B$ are distinct points on $\gamma$. Let $P$ and $Q$ be two points on $\gamma$. Lines $PA$ and $PB$ intersect line $l$ in two points $R$ and respectively $S$. Lines $QR$ and $QS$ intersect the second time circle $\gamma$ in points $C$ and $D$. Prove that the tangents from $C$ and $D$ to $\gamma$ are concurrent on line $l$.

Let $K,L,M$ be points on sides $AB,BC,CA$, respectively, of a triangle $ABC$ such that $AK/AB = BL/BC = CM/CA = 1/3$. Show that if the circumcircles of the triangles $AKM, BLK, CML$ are equal, then so are the incircles of these triangles.

Two segments $A_1B_1$ and $A_2B_2$ are given on the plane, with $\frac{A_2B_2}{A_1B_1} = k < 1$. On segment $A_1A_2$, point $A_3$ is taken, and on the extension of this segment beyond point $A_2$, point $A_4$ is taken, so $\frac{A_3A_2}{A_3A_1} =\frac{A_4A_2}{A_4A_1}= k$. Similarly, point $B_3$ is taken on segment $B_1B_2$ , and on the extension of this the segment beyond point $B_2$ is point $B_4$, so $\frac{B_3B_2}{B_3B_1} =\frac{B_4B_2}{B_4B_1}= k$. Find the angle between lines $A_3B_3$ and $A_4B_4$. (Netherlands)

In trapezoid $ABCD$ holds $AD \mid \mid BC$, $\angle ABC = 30^{\circ}$, $\angle BCD = 60^{\circ}$ and $BC=7$. Let $E$, $M$, $F$ and $N$ be midpoints of sides $AB$, $BC$, $CD$ and $DA$, respectively. If $MN=3$, find $EF$

In the triangle$ABC$ ($AC > AB$), point $N$ is the midpoint of $BC$, and $I$ is the intersection point of the angle bisectors. Ray $AI$ intersects the circumscribed circle of triangle $ABC$ at point $W$, a perpendicular $WF$ is drawn from it on side $AC$. Find the length of the segment $CF$ , if the radius of the circle inscribed in the triangle $ABC$ is equal to $r$ and $\angle INB = 45^o$. (Gryhoriy Filippovskyi)

The internal angle bisectors of $A$, $B$, and $C$ in $\triangle ABC$ concur at $I$ and intersect the circumcircle of $\triangle ABC$ at $L$, $M$, and $N$, respectively. The circle with diameter $IL$ intersects $BC$ at $D$ and $E$; the circle with diameter $IM$ intersects $CA$ at $F$ and $G$; the circle with diameter $IN$ intersects $AB$ at $H$ and $J$. Show that $D$, $E$, $F$, $G$, $H$, and $J$ are concyclic.

Point $I$ is incenter of triangle $ABC$ in which $AB \neq AC$. Lines $BI$ and $CI$ intersect sides $AC$ and $AB$ in points $D$ and $E$, respectively. Determine all measures of angle $BAC$, for which may be $DI = EI$.

Let $B$ and $C$ be fixed points, and let $A$ be a variable point such that $\angle BAC$ is fixed. The midpoints of $AB$ and $AC$ are $D$ and $E$ respectively, and $F,G$ are points such that $DF\perp AB$, $EG\perp AC$ and $BF$ and $CG$ are perpendicular to $BC$. Prove that $BF\cdot CG$ remains constant as $A$ varies.

In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$? $\textbf{(A) }6\sqrt{2}\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }9\sqrt{2}\qquad\textbf{(E) }32$ [asy] draw((-4,6*sqrt(2))--(4,6*sqrt(2))); draw((-4,-6*sqrt(2))--(4,-6*sqrt(2))); draw((-8,0)--(-4,6*sqrt(2))); draw((-8,0)--(-4,-6*sqrt(2))); draw((4,6*sqrt(2))--(8,0)); draw((8,0)--(4,-6*sqrt(2))); draw((-4,6*sqrt(2))--(4,6*sqrt(2))--(4,8+6*sqrt(2))--(-4,8+6*sqrt(2))--cycle); draw((-8,0)--(-4,-6*sqrt(2))--(-4-6*sqrt(2),-4-6*sqrt(2))--(-8-6*sqrt(2),-4)--cycle); label("$I$",(-4,8+6*sqrt(2)),dir(100)); label("$J$",(4,8+6*sqrt(2)),dir(80)); label("$A$",(-4,6*sqrt(2)),dir(280)); label("$B$",(4,6*sqrt(2)),dir(250)); label("$C$",(8,0),W); label("$D$",(4,-6*sqrt(2)),NW); label("$E$",(-4,-6*sqrt(2)),NE); label("$F$",(-8,0),E); draw((4,8+6*sqrt(2))--(4,6*sqrt(2))--(4+4*sqrt(3),4+6*sqrt(2))--cycle); label("$K$",(4+4*sqrt(3),4+6*sqrt(2)),E); draw((4+4*sqrt(3),4+6*sqrt(2))--(8,0),dashed); label("$H$",(-4-6*sqrt(2),-4-6*sqrt(2)),S); label("$G$",(-8-6*sqrt(2),-4),W); label("$32$",(-10,-8),N); label("$18$",(0,6*sqrt(2)+2),N); [/asy]

Constant points $B$ and $C$ lie on the circle $\omega$. The point middle of $BC$ is named $M$ by us. Assume that $A$ is a variable point on the $\omega$ and $H$ is the orthocenter of the triangle $ABC$. From the point $H$ we drop a perpendicular line to $MH$ to intersect the lines $AB$ and $AC$ at $X$ and $Y$ respectively. Prove that with the movement of $A$ on the $\omega$, the orthocenter of the triangle $AXY$ also moves on a circle.

In rectangle $ABCD$, point $M$ is the midpoint of $AB$ and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$. [i]Proposed by Michael Tang[/i]

Through the endpoints $A$ and $B$ of a diameter $AB$ of a given circle, the tangents $\ell$ and $m$ have been drawn. Let $C\ne A$ be a point on $\ell$ and let $q_1,q_2$ be two rays from $C$. Ray $q_i$ cuts the circle in $D_i$ and $E_i$ with $D_i$ between $C$ and $E_i, i = 1,2$. Rays $AD_1,AD_2,AE_1,AE_2$ meet $m$ in the respective points $M_1,M_2,N_1,N_2$. Prove that $M_1M_2 = N_1N_2$.

[u]Round 1[/u] [b]p1. [/b]The Queen of Bees invented a new language for her hive. The alphabet has only $6$ letters: A, C, E, N, R, T; however, the alphabetic order is different than in English. A word is any sequence of $6$ different letters. In the dictionary for this language, the word TRANCE immediately follows NECTAR. What is the last word in the dictionary? [b]p2.[/b] Is it possible to solve the equation $\frac{1}{x}= \frac{1}{y} +\frac{1}{z}$ with $x,y,z$ integers (positive or negative) such that one of the numbers $x,y,z$ has one digit, another has two digits, and the remaining one has three digits? [b]p3.[/b] The $10,000$ dots in a $100\times 100$ square grid are all colored blue. Rekha can paint some of them red, but there must always be a blue dot on the line segment between any two red dots. What is the largest number of dots she can color red? The picture shows a possible coloring for a $5\times 7$ grid. [img]https://cdn.artofproblemsolving.com/attachments/0/6/795f5ab879938ed2a4c8844092b873fb8589f8.jpg[/img] [b]p4.[/b] Six flies rest on a table. You have a swatter with a checkerboard pattern, much larger than the table. Show that there is always a way to position and orient the swatter to kill at least five of the flies. Each fly is much smaller than a swatter square and is killed if any portion of a black square hits any part of the fly. [b]p5.[/b] Maryam writes all the numbers $1-81$ in the cells of a $9\times 9$ table. Tian calculates the product of the numbers in each of the nine rows, and Olga calculates the product of the numbers in every column. Could Tian's and Olga's lists of nine products be identical? [u]Round 2[/u] [b]p6.[/b] A set of points in the plane is epic if, for every way of coloring the points red or blue, it is possible to draw two lines such that each blue point is on a line, but none of the red points are. The figure shows a particular set of $4$ points and demonstrates that it is epic. What is the maximum possible size of an epic set? [img]https://cdn.artofproblemsolving.com/attachments/e/f/44fd1679c520bdc55c78603190409222d0b721.jpg[/img] [b]p7.[/b] Froggy Chess is a game played on a pond with lily pads. First Judit places a frog on a pad of her choice, then Magnus places a frog on a different pad of his choice. After that, they alternate turns, with Judit moving first. Each player, on his or her turn, selects either of the two frogs and another lily pad where that frog must jump. The jump must reduce the distance between the frogs (all distances between the lily pads are different), but both frogs cannot end up on the same lily pad. Whoever cannot make a move loses. The picture below shows the jumps permitted in a particular situation. Who wins the game if there are $2017$ lily pads? [img]https://cdn.artofproblemsolving.com/attachments/a/9/1a26e046a2a614a663f9d317363aac61654684.jpg[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that the sum of the plane angles at each of the vertices of a given tetrahedron is $ 180^{\circ} $ if and only if all its faces are congruent.

In a triangle $ABC$, $AB = AC$, and $D$ is on $BC$. A point $E$ is chosen on $AC$, and a point $F$ is chosen on $AB$, such that $DE = DC$ and $DF = DB$. It is given that $\frac{DC}{BD} = 2$ and $\frac{AF}{AE} = 5$. Determine that value of $\frac{AB}{BC}$.

Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle. [b]a.)[/b] Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle. [b]b.)[/b] Show that generally, one gets two possible points $P$ ($P_{1}$ and $P_{2}$) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.

[u]Round 5[/u] [b]p13.[/b] Five different schools are competing in a tournament where each pair of teams plays at most once. Four pairs of teams are randomly selected and play against each other. After these four matches, what is the probability that Chad's and Jordan's respective schools have played against each other, assuming that Chad and Jordan come from different schools? [b]p14.[/b] A square of side length $1$ and a regular hexagon are both circumscribed by the same circle. What is the side length of the hexagon? [b]p15.[/b] From the list of integers $1,2, 3,...,30$ Jordan can pick at least one pair of distinct numbers such that none of the $28$ other numbers are equal to the sum or the difference of this pair. Of all possible such pairs, Jordan chooses the pair with the least sum. Which two numbers does Jordan pick? [u]Round 6[/u] [b]p16.[/b] What is the sum of all two-digit integers with no digit greater than four whose squares also have no digit greater than four? [b]p17.[/b] Chad marks off ten points on a circle. Then, Jordan draws five chords under the following constraints: $\bullet$ Each of the ten points is on exactly one chord. $\bullet$ No two chords intersect. $\bullet$ There do not exist (potentially non-consecutive) points $A, B,C,D,E$, and $F$, in that order around the circle, for which $AB$, $CD$, and $EF$ are all drawn chords. In how many ways can Jordan draw these chords? [b]p18.[/b] Chad is thirsty. He has $109$ cubic centimeters of silicon and a 3D printer with which he can print a cup to drink water in. He wants a silicon cup whose exterior is cubical, with five square faces and an open top, that can hold exactly $234$ cubic centimeters of water when filled to the rim in a rectangular-box-shaped cavity. Using all of his silicon, he prints a such cup whose thickness is the same on the five faces. What is this thickness, in centimeters? [u]Round 7[/u] [b]p19.[/b] Jordan wants to create an equiangular octagon whose side lengths are exactly the first $8$ positive integers, so that each side has a different length. How many such octagons can Jordan create? [b]p20.[/b] There are two positive integers on the blackboard. Chad computes the sum of these two numbers and tells it to Jordan. Jordan then calculates the sum of the greatest common divisor and the least common multiple of the two numbers, and discovers that her result is exactly $3$ times as large as the number Chad told her. What is the smallest possible sum that Chad could have said? [b]p21.[/b] Chad uses yater to measure distances, and knows the conversion factor from yaters to meters precisely. When Jordan asks Chad to convert yaters into meters, Chad only gives Jordan the result rounded to the nearest integer meters. At Jordan's request, Chad converts $5$ yaters into $8$ meters and $7$ yaters into $12$ meters. Given this information, how many possible numbers of meters could Jordan receive from Chad when requesting to convert $2014$ yaters into meters? [u]Round 8[/u] [b]p22.[/b] Jordan places a rectangle inside a triangle with side lengths $13$, $14$, and $15$ so that the vertices of the rectangle all lie on sides of the triangle. What is the maximum possible area of Jordan's rectangle? [b]p23.[/b] Hoping to join Chad and Jordan in the Exeter Space Station, there are $2014$ prospective astronauts of various nationalities. It is given that $1006$ of the astronaut applicants are American and that there are a total of $64$ countries represented among the applicants. The applicants are to group into $1007$ pairs with no pair consisting of two applicants of the same nationality. Over all possible distributions of nationalities, what is the maximum number of possible ways to make the $1007$ pairs of applicants? Express your answer in the form $a \cdot b!$, where $a$ and $b$ are positive integers and $a$ is not divisible by $b + 1$. Note: The expression $k!$ denotes the product $k \cdot (k - 1) \cdot ... \cdot 2 \cdot 1$. [b]p24.[/b] We say a polynomial $P$ in $x$ and $y$ is $n$-[i]good [/i] if $P(x, y) = 0$ for all integers $x$ and $y$, with $x \ne y$, between $1$ and $n$, inclusive. We also define the complexity of a polynomial to be the maximum sum of exponents of $x$ and $y$ across its terms with nonzero coeffcients. What is the minimal complexity of a nonzero $4$-good polynomial? In addition, give an example of a $4$-good polynomial attaining this minimal complexity. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2915803p26040550]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\mathcal S$ be a set of $16$ points in the plane, no three collinear. Let $\chi(S)$ denote the number of ways to draw $8$ lines with endpoints in $\mathcal S$, such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of $\chi(\mathcal S)$ across all such $\mathcal S$. [i]Ankan Bhattacharya[/i]

Let $B$ be a bounded closed convex symmetric (with respect to the origin) set in $\mathbb{R}^{2}$ with boundary $\Gamma$. Let $B$ have the property that the ellipse of maximal area contained in $B$ is the disc $D$ of radius $1$ centered at the origin with boundary $C$. Prove that $A \cap \Gamma \ne \emptyset$ for any arc $A$ of $C$ of length $l(A)\geq \frac{\pi}{2}$.