Let $ABCD$ be a cyclic quadrilateral and let $E$ and $F$ be the points on the sides $AB$ and $CD$ respectively such that $AE : EB = CF : FD$. Point $P$ on the segment EF satsfies $EP : PF = AB : CD$. Prove that the ratio of the areas of $\vartriangle APD$ and $\vartriangle BPC$ does not depend on the choice of $E$ and $F$.

Trapezoid $ ABCD$, with $ \overline{AB}\parallel{}\overline{CD}$, is inscribed in circle $ \omega$ and point $ G$ lies inside triangle $ BCD$. Rays $ AG$ and $ BG$ meet $ \omega$ again at points $ P$ and $ Q$, respectively. Let the line through $ G$ parallel to $ \overline{AB}$ intersects $ \overline{BD}$ and $ \overline{BC}$ at points $ R$ and $ S$, respectively. Prove that quadrilateral $ PQRS$ is cyclic if and only if $ \overline{BG}$ bisects $ \angle CBD$.

Let $ ABCD$ be a convex quadrilateral and let $ O \in AC \cap BD$, $ P \in AB \cap CD$, $ Q \in BC \cap DA$. If $ R$ is the orthogonal projection of $ O$ on the line $ PQ$ prove that the orthogonal projections of $ R$ on the sidelines of $ ABCD$ are concyclic.

In triangle $\triangle ABC$, the points $A', B', C'$ are on sides $BC, AC, AB$ respectively. Also, $AA', BB', CC'$ intersect at the point $O$(they are concurrent at $O$). Also, $\frac {AO}{OA'}+\frac {BO}{OB'}+\frac {CO}{OC'} = 92$. Find the value of $\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}$.

In a triangle $ABC$ let $A_1$ be the midpoint of side $BC$. Draw circles with centers $A, A1$ and radii $AA_1, BC$ respectively and let $A'A''$ be their common chord. Similarly denote the segments $B'B''$ and $C'C''$. Show that lines $A'A'', B'B'''$ and $C'C''$ are concurrent.

An acute triangle $ ABC$ has three heights $ AD, BE$ and $ CF$ respectively. Prove that the perimeter of triangle $ DEF$ is not over half of the perimeter of triangle $ ABC.$

[u]Set 1[/u] [b]p1.[/b] Find $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11$. [b]p2.[/b] Find $1 \cdot 11 + 2 \cdot 10 + 3 \cdot 9 + 4 \cdot 8 + 5 \cdot 7 + 6 \cdot 6$. [b]p3.[/b] Let $\frac{1}{1\cdot 2} +\frac{1}{2\cdot 3} +\frac{1}{3\cdot 4} +\frac{1}{4\cdot 5} +\frac{1}{5\cdot 6} +\frac{1}{6\cdot 7} +\frac{1}{7\cdot 8} +\frac{1}{8\cdot 9} +\frac{1}{9\cdot 10} +\frac{1}{10\cdot 11} =\frac{m}{n}$ , where $m$ and $n$ are positive integers that share no prime divisors. Find $m + n$. [u]Set 2[/u] [b]p4.[/b] Define $0! = 1$ and let $n! = n \cdot (n - 1)!$ for all positive integers $n$. Find the value of $(2! + 0!)(1! + 8!)$. [b]p5.[/b] Rachel’s favorite number is a positive integer $n$. She gives Justin three clues about it: $\bullet$ $n$ is prime. $\bullet$ $n^2 - 5n + 6 \ne 0$. $\bullet$ $n$ is a divisor of $252$. What is Rachel’s favorite number? [b]p6.[/b] Shen eats eleven blueberries on Monday. Each day after that, he eats five more blueberries than the day before. For example, Shen eats sixteen blueberries on Tuesday. How many blueberries has Shen eaten in total before he eats on the subsequent Monday? [u]Set 3[/u] [b]p7.[/b] Triangle $ABC$ satisfies $AB = 7$, $BC = 12$, and $CA = 13$. If the area of $ABC$ can be expressed in the form $m\sqrt{n}$, where $n$ is not divisible by the square of a prime, then determine $m + n$. [b]p8.[/b] Sebastian is playing the game Split! on a coordinate plane. He begins the game with one token at $(0, 0)$. For each move, he is allowed to select a token on any point $(x, y)$ and take it off the plane, replacing it with two tokens, one at $(x + 1, y)$, and one at $(x, y + 1)$. At the end of the game, for a token on $(a, b)$, it is assigned a score $\frac{1}{2^{a+b}}$ . These scores are summed for his total score. Determine the highest total score Sebastian can get in $100$ moves. [b]p9.[/b] Find the number of positive integers $n$ satisfying the following two properties: $\bullet$ $n$ has either four or five digits, where leading zeros are not permitted, $\bullet$ The sum of the digits of $n$ is a multiple of $3$. [u]Set 4[/u] [b]p10.[/b] [i]A unit square rotated $45^o$ about a vertex, Sweeps the area for Farmer Khiem’s pen. If $n$ is the space the pigs can roam, Determine the floor of $100n$.[/i] If $n$ is the area a unit square sweeps out when rotated 4$5$ degrees about a vertex, determine $\lfloor 100n \rfloor$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/1/129efd0dbd56dc0b4fb742ac80eaf2447e106d.png[/img] [b]p11.[/b][i] Michael is planting four trees, In a grid, three rows of three, If two trees are close, Then both are bulldozed, So how many ways can it be?[/i] In a three by three grid of squares, determine the number of ways to select four squares such that no two share a side. [b]p12.[/b] [i]Three sixty-seven Are the last three digits of $n$ cubed. What is $n$?[/i] If the last three digits of $n^3$ are $367$ for a positive integer $n$ less than $1000$, determine $n$. [u]Set 5[/u] [b]p13.[/b] Determine $\sqrt[4]{97 + 56\sqrt{3}} + \sqrt[4]{97 - 56\sqrt{3}}$. [b]p14. [/b]Triangle $\vartriangle ABC$ is inscribed in a circle $\omega$ of radius $12$ so that $\angle B = 68^o$ and $\angle C = 64^o$ . The perpendicular from $A$ to $BC$ intersects $\omega$ at $D$, and the angle bisector of $\angle B$ intersects $\omega$ at $E$. What is the value of $DE^2$? [b]p15.[/b] Determine the sum of all positive integers $n$ such that $4n^4 + 1$ is prime. [u]Set 6[/u] [b]p16.[/b] Suppose that $p, q, r$ are primes such that $pqr = 11(p + q + r)$ such that $p\ge q \ge r$. Determine the sum of all possible values of $p$. [b]p17.[/b] Let the operation $\oplus$ satisfy $a \oplus b =\frac{1}{1/a+1/b}$ . Suppose $$N = (...((2 \oplus 2) \oplus 2) \oplus ... 2),$$ where there are $2018$ instances of $\oplus$ . If $N$ can be expressed in the form $m/n$, where $m$ and $n$ are relatively prime positive integers, then determine $m + n$. [b]p18.[/b] What is the remainder when $\frac{2018^{1001} - 1}{2017}$ is divided by $2017$? PS. You had better use hide for answers. Last sets have been posted [url=https://artofproblemsolving.com/community/c4h2777307p24369763]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We call a pair of polygons, $p$ and $q$, [i]nesting[/i] if we can draw one inside the other, possibly after rotation and/or reflection; otherwise we call them [i]non-nesting[/i]. Let $p$ and $q$ be polygons. Prove that if we can find a polygon $r$, which is similar to $q$, such that $r$ and $p$ are non-nesting if and only if $p$ and $q$ are not similar.

Let $\vartriangle ABC$ be an acute triangle with altitudes $AD$, $BE$, $CF$ and orthocenter $H$. Circle $\odot V$ is the circumcircle of $\vartriangle DE F$. Let segments $FD$, $BH$ intersect at point $P$. Let segments $ED$, $HC$ intersect at point $Q$. Let $K$ be a point on $AC$ such that $VK \perp CF$. a) Prove that $\vartriangle PQH \sim \vartriangle AKV$. b) Let line $PQ$ intersect $\odot V$ at points $I,G$. Prove that points $B,I,H,G,C$ are concyclic [hide]with center the symmetric point $X$ of circumcenter $O$ of $\vartriangle ABC$ wrt $BC$.[/hide] [hide=PS.] There is a chance that those in the hide were not wanted in the problem, as I tried to understand the wording from a solutions' video. I couldn't find the original wording pdf or picture.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/c/3/0b934c5756461ff854d38f51ef4f76d55cbd95.png[/img] [url=https://www.geogebra.org/m/cjduebke]geogebra file[/url]

Let $ABC$ be a triangle such that $AB < AC$. Let the excircle opposite to A be tangent to the lines $AB, AC$, and $BC$ at points $D, E$, and $F$, respectively, and let $J$ be its centre. Let $P$ be a point on the side $BC$. The circumcircles of the triangles $BDP$ and $CEP$ intersect for the second time at $Q$. Let $R$ be the foot of the perpendicular from $A$ to the line $FJ$. Prove that the points $P, Q$, and $R$ are collinear. (The [i]excircle[/i] of a triangle $ABC$ opposite to $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.) [i]Proposed by Bozhidar Dimitrov, Bulgaria[/i]

The hexagon $ABLCDK$ is inscribed and the line $LK$ intersects the segments $AD, BC, AC$ and $BD$ in points $M, N, P$ and $Q$, respectively. Prove that $NL \cdot KP \cdot MQ = KM \cdot PN \cdot LQ$.

In circle $ O$, the midpoint of radius $ OX$ is $ Q$; at $ Q$, $ \overline{AB} \perp \overline{XY}$. The semi-circle with $ \overline{AB}$ as diameter intersects $ \overline{XY}$ in $ M$. Line $ \overline{AM}$ intersects circle $ O$ in $ C$, and line $ \overline{BM}$ intersects circle $ O$ in $ D$. Line $ \overline{AD}$ is drawn. Then, if the radius of circle $ O$ is $ r$, $ AD$ is: [asy]defaultpen(linewidth(.8pt)); unitsize(2.5cm); real m = 0; real b = 0; pair O = origin; pair X = (-1,0); pair Y = (1,0); pair Q = midpoint(O--X); pair A = (Q.x, -1*sqrt(3)/2); pair B = (Q.x, -1*A.y); pair M = (Q.x + sqrt(3)/2,0); m = (B.y - M.y)/(B.x - M.x); b = (B.y - m*B.x); pair D = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b)); m = (A.y - M.y)/(A.x - M.x); b = (A.y - m*A.x); pair C = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b)); draw(Circle(O,1)); draw(Arc(Q,sqrt(3)/2,-90,90)); draw(A--B); draw(X--Y); draw(B--D); draw(A--C); draw(A--D); dot(O);dot(M); label("$B$",B,NW); label("$C$",C,NE); label("$Y$",Y,E); label("$D$",D,SE); label("$A$",A,SW); label("$X$",X,W); label("$Q$",Q,SW); label("$O$",O,SW); label("$M$",M,NE+2N);[/asy]$ \textbf{(A)}\ r\sqrt {2} \qquad \textbf{(B)}\ r\qquad \textbf{(C)}\ \text{not a side of an inscribed regular polygon}\qquad \textbf{(D)}\ \frac {r\sqrt {3}}{2}\qquad \textbf{(E)}\ r\sqrt {3}$

Let $ABC$ be a scalene (i.e. non-isosceles) triangle. Let $U$ be the center of the circumcircle of this triangle and $I$ the center of the incircle. Assume that the second point of intersection different from $C$ of the angle bisector of $\gamma = \angle ACB$ with the circumcircle of $ABC$ lies on the perpendicular bisector of $UI$. Show that $\gamma$ is the second-largest angle in the triangle $ABC$.

A point $O$ is choosen inside a triangle $ABC$ so that the length of segments $OA$, $OB$ and $OC$ are equal to $15$,$12$ and $20$, respectively. It is known that the feet of the perpendiculars from $O$ to the sides of the triangle $ABC$ are the vertices of an equilateral triangle. Find the value of the angle $BAC$.

$\bigtriangleup ABC$ triangle such that $AB = AC, \angle BAC = 20 \textdegree$. $P$ is on $AB$ such that $AP = BC$, find $\frac{1}{2}\angle APC$ in degrees.

Let $D,E,F$ be points on the sides $BC,CA,AB$ of a triangle $ABC$, respectively. Suppose $AD, BE,CF$ are concurrent at $P$. If $PF/PC =2/3, PE/PB = 2/7$ and $PD/PA = m/n$, where $m, n$ are positive integers with $gcd(m, n) = 1$, find $m + n$.

There is three-dimensional space. For every integer $n$ we build planes $ x \pm y\pm z = n$. All space is divided on octahedrons and tetrahedrons. Point $(x_0,y_0,z_0)$ has rational coordinates but not lies on any plane. Prove, that there is such natural $k$ , that point $(kx_0,ky_0,kz_0)$ lies strictly inside the octahedron of partition.

Quadrilateral $ABCD$ is circumscribed, rays $BA$ and $CD$ intersect in point $E$, rays $BC$ and $AD$ intersect in point $F$. The incircle of the triangle formed by lines $AB, CD$ and the bisector of angle $B$, touches $AB$ in point $K$, and the incircle of the triangle formed by lines $AD, BC$ and the bisector of angle $B$, touches $BC$ in point $L$. Prove that lines $KL, AC$ and $EF$ concur. (I.Bogdanov)

Let $ A,B,C,M$ points in the plane and no three of them are on a line. And let $ A',B',C'$ points such that $ MAC'B, MBA'C$ and $ MCB'A$ are parallelograms: (a) Show that \[ \overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} < \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.\] (b) Assume segments $ AA', BB'$ and $ CC'$ have the same length. Show that $ 2 \left(\overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} \right) \leq \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.$ When do we have equality?

In an equilateral triangle $ABC$, let $D$ be a point on the side $AB$ such that $AD = AB /n$. Prove that the sum of $n - 1$ angles $\angle DP_lA$, $\angle DP_2A$, $...$, $\angle DP_nA$ where $P_1$, $P_2$, $...$ ,$P_{n-1}$ are the points dividing the side $BC$ into $n$ equal parts, is equal to $30$ degrees if (a) $n = 3$ (b) $n$ is an arbitrary integer, $n > 2$. (V Proizvolov)

In an acute triangle $ABC$, let $M$ and $N$ be the midpoints of $AB$ and $AC$ and let $BH$ be its altitude from $B$. Its incircle touches $AC$ at $K$ and the line through $K$ parallel to $MH$ meets $MN$ at $P$. Prove that $AMPK$ has an incircle.

Let $X,Y$ be two points which lies on the line $BC$ of $\triangle ABC(X,B,C,Y\text{lies in sequence})$ such that $BX\cdot AC=CY\cdot AB$, $O_1,O_2$ are the circumcenters of $\triangle ACX,\triangle ABY$, $O_1O_2\cap AB=U,O_1O_2\cap AC=V$. Prove that $\triangle AUV$ is a isosceles triangle.

It is given a circle with center $O$ and radius $r$. $AB$ and $MN$ are two diameters. The lines $MB$ and $NB$ are tangent to the circle at the points $M'$ and $N'$ and intersect at point $A$. $M''$ and $N''$ are the midpoints of the segments $AM'$ and $AN'$. Prove that: (a) the points $M,N,N',M'$ are concyclic. (b) the heights of the triangle $M''N''B$ intersect in the midpoint of the radius $OA$.

In triangle $ABC$ the incenter is $I$. The center of the excircle opposite $A$ is $I_A$, and it is tangent to $BC$ at $D$. The midpoint of arc $BAC$ is $N$, and $NI$ intersects $(ABC)$ again at $T$. The center of $(AID)$ is $K$. Prove that $TI_A\perp KI$.

Given trapezoid $ABCD$ with parallel sides $AB$ and $CD$, let $E$ be a point on line $BC$ outside segment $BC$, such that segment $AE$ intersects segment $CD$. Assume that there exists a point $F$ inside segment $AD$ such that $\angle EAD=\angle CBF$. Denote by $I$ the point of intersection of $CD$ and $EF$, and by $J$ the point of intersection of $AB$ and $EF$. Let $K$ be the midpoint of segment $EF$, and assume that $K$ is different from $I$ and $J$. Prove that $K$ belongs to the circumcircle of $\triangle ABI$ if and only if $K$ belongs to the circumcircle of $\triangle CDJ$.