Let $A$ be a fixed point of a circle $\omega$. Let $BC$ be an arbitrary chord of $\omega$ passing through a fixed point $P$. Prove that the nine-points circles of triangles $ABC$ touch some fixed circle not depending on $BC$.

The set $ G$ is defined by the points $ (x,y)$ with integer coordinates, $ 3\le|x|\le7$, $ 3\le|y|\le7$. How many squares of side at least $ 6$ have their four vertices in $ G$? [asy]defaultpen(black+0.75bp+fontsize(8pt)); size(5cm); path p = scale(.15)*unitcircle; draw((-8,0)--(8.5,0),Arrow(HookHead,1mm)); draw((0,-8)--(0,8.5),Arrow(HookHead,1mm)); int i,j; for (i=-7;i<8;++i) { for (j=-7;j<8;++j) { if (((-7 <= i) && (i <= -3)) || ((3 <= i) && (i<= 7))) { if (((-7 <= j) && (j <= -3)) || ((3 <= j) && (j<= 7))) { fill(shift(i,j)*p,black); }}}} draw((-7,-.2)--(-7,.2),black+0.5bp); draw((-3,-.2)--(-3,.2),black+0.5bp); draw((3,-.2)--(3,.2),black+0.5bp); draw((7,-.2)--(7,.2),black+0.5bp); draw((-.2,-7)--(.2,-7),black+0.5bp); draw((-.2,-3)--(.2,-3),black+0.5bp); draw((-.2,3)--(.2,3),black+0.5bp); draw((-.2,7)--(.2,7),black+0.5bp); label("$-7$",(-7,0),S); label("$-3$",(-3,0),S); label("$3$",(3,0),S); label("$7$",(7,0),S); label("$-7$",(0,-7),W); label("$-3$",(0,-3),W); label("$3$",(0,3),W); label("$7$",(0,7),W);[/asy]$ \textbf{(A)}\ 125\qquad \textbf{(B)}\ 150\qquad \textbf{(C)}\ 175\qquad \textbf{(D)}\ 200\qquad \textbf{(E)}\ 225$

Consider the circle $\gamma$ with center at point $(0,3)$ and radius $3$, and a line $r$ parallel to the axis $Ox$ at a distance $3$ from the origin. A variable line through the origin meets $\gamma$ at point $M$ and $r$ at point $P$. Find the locus of the intersection point of the lines through $M$ and $P$ parallel to $Ox$ and $Oy$ respectively.

Consider an isosceles triangle $ABC$ ($AB = BC$). Let $D$ be on $BC$ such that $AD \perp BC$ and $O$ be a circle with diameter $BC$. Suppose that segment $AD$ intersects circle $O$ at $E$. If $CA = 2$ what is $CE$?

Let $ABCD$ be a convex quadrilateral, and the points $A_1 \in (CD)$, $A_2 \in (BC)$, $C_1 \in (AB)$, $C_2 \in (AD)$. Let $M, N$ be the intersection points between the lines $AA_2, CC_1$ and $AA_1, CC_2$ respectively. Prove that if three of the quadrilaterals $ABCD$, $A_2BC_1M$, $AMCN$, $A_1NC_2D$ are circumscriptive (i.e. there exists an incircle tangent to all the sides of the quadrilateral) then the forth quadrilateral is also circumscriptive.

The incircle of a triangle $ABC$ touches its sides $BC$, $CA$, $AB$ at the points $X$, $Y$, $Z$, respectively. Let $X^{\prime}$, $Y^{\prime}$, $Z^{\prime}$ be the reflections of these points $X$, $Y$, $Z$ in the external angle bisectors of the angles $CAB$, $ABC$, $BCA$, respectively. Show that $Y^{\prime}Z^{\prime}\parallel BC$, $Z^{\prime}X^{\prime}\parallel CA$ and $X^{\prime}Y^{\prime}\parallel AB$.

Points $A, B, C,$ and $D{}$ lie on the surface of a sphere with diameter 1. Determine the maximum possible volume of tetrahedron $ABCD.$ [i]Proposed by Fredy Yip[/i]

Points $A,B$ lies on the circle $S$. Tangent lines to $S$ at $A$ and $B$ intersects at $C$. $M$ -midpoint of $AB$. Circle $S_1$ goes through $M,C$ and intersects $AB$ at $D$ and $S$ at $K$ and $L$. Prove, that tangent lines to $S$ at $K$ and $L$ intersects at point on the segment $CD$.

Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in $BC$, and $l_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B$, $I'_B$ and line $l_B$ analogously. Let $P$ be the intersection point of $l_A$ and $l_B$. [list=a] [*] Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$. [*] Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle XIY = 120^{\circ}$. [/list]

Let $ABC$ be an acute angled triangle and let $D$ be an interior point of the segment $BC$. Let the circumcircle of $ACD$ intersect $AB$ at $E$ ($E$ between $A$ and $B$) and let circumcircle of $ABD$ intersect $AC$ at $F$ ($F$ between $A$ and $C$). Let $O$ be the circumcenter of $AEF$. Prove that $OD$ bisects $\angle EDF$.

The plane is divided into unit cells, and each of the cells is painted in one of two given colors. Find the minimum possible number of cells in a figure consisting of entire cells which contains each of the $16$ possible colored $2\times2$ squares.

Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.

In triangle $ABC$, $AB=13$, $BC=15$, and $CA=14$. Point $D$ is on $\overline{BC}$ with $CD=6.$ Point $E$ is on $\overline{BC}$ such that $\angle BAE\cong \angle CAD.$ Given that $BE=\frac pq$ where $p$ and $q$ are relatively prime positive integers, find $q.$

[u]Round 5[/u] [b]p13.[/b] Given that $x^3 + y^3 = 208$ and $x + y = 4$, what is the value of $\frac{1}{x} +\frac{1}{y}$? [b]p14.[/b] Find the sum of all three-digit integers $n$ such that the value of $n$ is equal to the sum of the factorials of $n$’s digits. [b]p15.[/b] Three christmas lights are initially off. The Grinch decides to fiddle around with the lights, switching one of the lights each second. He wishes to get every possible combination of lights. After how many seconds can the Grinch complete his task? [u]Round 6[/u] [b]p16.[/b] A regular tetrahedron of side length $1$ has four similar tetrahedrons of side length $1/2$ chopped off, one from each of the four vertices. What is the sum of the numbers of vertices, edges, and faces of the remaining solid? [b]p17.[/b] Mario serves a pie in the shape of a regular $2013$-gon. To make each slice, he must cut in a straight line starting from one vertex and ending at another vertex of the pie. Every vertex of a slice must be a vertex of the original $2013$-gon. If every person eats at least one slice of pie regardless of the size, what is the maximum number of people the $2013$-gon pie can feed? [b]p18.[/b] Find the largest integer $x$ such that $x^2 + 1$ divides $x^3 + x - 1000$. [u]Round 7[/u] [b]p19.[/b] In $\vartriangle ABC$, $\angle B = 87^o$, $\angle C = 29^o$, and $AC = 37$. The perpendicular bisector of $\overline{BC}$ meets $\overline{AC}$ at point $T$. What is the value of $AB + BT$? [b]p20.[/b] Consider the sequence $f(1) = 1$, $f(2) = \frac12$ ,$ f(3) =\frac{1+3}{2}$, $f(4) =\frac{ 1+3}{2+4}$ ,$ f(5) = \frac{ 1+3+5}{2+4} . . . $ What is the minimum value of $n$, with $n > 1$, such that $|f(n) - 1| \le \frac{1}{10 }$. [b]p21.[/b] Three unit circles are centered at $(0, 0)$,$(0, 2)$, and $(2, 0)$. A line is drawn passing through $(0, 1)$ such that the region inside the circles and above the line has the same area as the region inside the circles and below the line. What is the equation of this line in $y = mx + b$ form? [u]Round 8[/u] [b]p22.[/b] The two walls of a pinball machine are positioned at a $45$ degree angle to each other. A pinball, represented by a point, is fired at a wall (but not at the intersection of the two walls). What is the maximum number of times the ball can bounce off the walls? [b]p23.[/b] Albert is fooling people with his weighted coin at a carnival. He asks his guests to guess how many times heads will show up if he flips the coin $4$ times. Richard decides to play the game and guesses that heads will show up $2$ times. In the previous game, Zach guessed that the heads would show up 3 times. In Zach’s game, there were least 3 heads, and given this information, Zach had a $\frac49$ chance of winning. What is the probability that Richard guessescorrectly? [b]p24.[/b] Let $S$ be the set of all positive integers relatively prime to $2013$ that have no prime factor greater than $15$. Find the sum of the reciprocals of all of the elements of $S$. PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134546p28406927]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3137069p28442224]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two points $K$ and $L$ are chosen inside triangle $ABC$ and a point $D$ is chosen on the side $AB$. Suppose that $B$, $K$, $L$, $C$ are concyclic, $\angle AKD = \angle BCK$ and $\angle ALD = \angle BCL$. Prove that $AK = AL$.

A set of points in the plane is called disharmonic, if the ratio of any two distances between the points is between $100/101$ and $101/100$, or at least $100$ or at most $1/100$. Is it true that for any distinct points $A_1,A_2,\ldots,A_n$ in the plane it is always possible to find distinct points $A_1',A_2',\ldots, A_n'$ that form a disharmonic set of points, and moreover $A_i, A_j$ and $A_k$ are collinear in this order if and only if $A_i', A_j'$ and $A_k'$ are collinear in this order (for all distinct $1 \le i,j,k\le n$? [i]Submitted by Dömötör Pálvölgyi and Balázs Keszegh, Budapest[/i]

Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

Some of the $n + 1$ cities in a country (including the capital city) are connected by one-way or two-way airlines. No two cities are connected by both a one-way airline and a two-way airline, but there may be more than one two-way airline between two cities. If $d_A$ denotes the number of airlines from a city $A$, then $d_A \le n$ for any city $A$ other than the capital city and $d_A + d_B \le n$ for any two cities $A$ and $B$ other than the capital city which are not connected by a two-way airline. Every airline has a return, possibly consisting of several connected flights. Find the largest possible number of two-way airlines and all configurations of airlines for which this largest number is attained.

Does there exist a triangle in which two sides are integer multiples of the median to that side? Does there exist a triangle in which every side is an integer multiple of the median to that side?

Let $I, O, \omega, \Omega$ be the incenter, circumcenter, the incircle, and the circumcircle, respectively, of a scalene triangle $ABC$. The incircle $\omega$ is tangent to side $BC$ at point $D$. Let $S$ be the point on the circumcircle $\Omega$ such that $AS, OI, BC$ are concurrent. Let $H$ be the orthocenter of triangle $BIC$. Point $T$ lies on $\Omega$ such that $\angle ATI$ is a right angle. Prove that the points $D, T, H, S$ are concyclic. [i]Proposed by ltf0501[/i]

Let $ABC$ be a triangle. Let $ r$ denote the inradius of $\vartriangle ABC$. Let $r_a$ denote the $A$-exradius of $\vartriangle ABC$. Note that the $A$-excircle of $\vartriangle ABC$ is the circle that is tangent to segment $BC$, the extension of ray $AB$ beyond $ B$ and the extension of $AC$ beyond $C$. The $A$-exradius is the radius of the $A$-excircle. Define $ r_b$ and $ r_c$ analogously. Prove that $$\frac{1}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}$$

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle. [i]Evan O'Dorney.[/i]

Determine the sum of all possible surface area of a cube two of whose vertices are $(1,2,0)$ and $(3,3,2)$.

Let $ABCD$ be a quadrilateral inscribed a circle $(O)$. Assume that $AB$ and $CD$ intersect at $E, AC$ and $BD$ intersect at $K$, and $O$ does not belong to the line $KE$. Let $G$ and $H$ be the midpoints of $AB$ and $CD$ respectively. Let $(I)$ be the circumcircle of the triangle $GKH$. Let $(I)$ and $(O)$ intersect at $M, N$ such that $MGHN$ is convex quadrilateral. Let $P$ be the intersection of $MG$ and $HN,Q$ be the intersection of $MN$ and $GH$. a) Prove that $IK$ and $OE$ are parallel. b) Prove that $PK$ is perpendicular to $IQ$.

Let $ABCD$ be a trapezium with $AB\parallel CD$. Let $P$ be a point on $AC$ such that $C$ is between $A$ and $P$; and let $X, Y$ be the midpoints of $AB, CD$ respectively. Let $PX$ intersect $BC$ in $N$ and $PY$ intersect $AD$ in $M$. Prove that $MN\parallel AB$.