Given a triangle $ ABC $, $ AD $ is its angle bisector. Let $ E, F $ be the centers of the circles inscribed in the triangles $ ADC $ and $ ADB $, respectively. Denote by $ \omega $, the circle circumscribed around the triangle $ DEF $, and by $ Q $, the intersection point of $ BE $ and $ CF $, and $ H, J, K, M $ , respectively the second intersection point of the lines $ CE, CF, BE, BF $ with circle $ \omega $. Let $\omega_1, \omega_2 $ the circles be circumscribed around the triangles $ HQJ $ and $ KQM $ Prove that the intersection point of the circles $\omega_1, \omega_2 $ different from $ Q $ lies on the line $ AD $. (Kivva Bogdan)

[b]p1.[/b] In their class Introduction to Ladders at Greendale Community College, Jan takes four tests. They realize that their test scores in chronological order form a strictly increasing arithmetic progression with integer terms, and that the average of those scores is an integer greater than or equal to $94$. How many possible combinations of test scores could they have had? (Test scores at Greendale range between $0$ and $100$, inclusive.) [b]p2.[/b] Suppose that $A$ and $B$ are digits between $1$ and $9$ such that $$0.\overline{ABABAB...}+ B \cdot (0.\overline{AAA...}) = A \cdot (0.\overline{B1B1B1...}) + 1$$ Find the sum of all possible values of $10A + B$. [b]p3.[/b] Let $ABC$ be an isosceles right triangle with $m\angle ABC = 90^o$. Let $D$ and $E$ lie on segments $\overline{AC}$ and $\overline{BC}$, respectively, such that triangles $\vartriangle ADB$ and $\vartriangle CDE$ are similar and $DE =EB$. If $\frac{AC}{AD} = 1 +\frac{\sqrt{a}}{b}$ with $a$, $b$ positive integers and $a$ squarefree, then find $a + b$. [b]p4.[/b] Five bowling pins $P_1, P_2, ..., P_5$ are lined up in a row. Each turn, Jemma picks a pin at random from the standing pins, and throws a bowling ball at that pin; that pin and each pin directly adjacent to it are knocked down. If the expected value of the number of turns Jemma will take to knock down all the pins is $\frac{a}{b}$ where $a$ and $b$ are relatively prime, find $a + b$. (Pins $P_i$ and $P_j$ are adjacent if and only if $|i - j| = 1$.) [b]p5.[/b] How many terms in the expansion of $$(1 + x + x^2 + x^3 +... + x^{2021})(1 + x^2 + x^4 + x^6 + ... + x^{4042})$$ have coeffcients equal to $1011$? [b]p6.[/b] Suppose $f(x)$ is a monic quadratic polynomial with distinct nonzero roots $p$ and $q$, and suppose $g(x)$ is a monic quadratic polynomial with roots $p + \frac{1}{q}$ and $q + \frac{1}{p}$ . If we are given that $g(-1) = 1$ and $f(0)\ne -1$, then there exists some real number $r$ that must be a root of $f(x)$. Find $r$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given is a circular bus route with $n\geqslant2$ bus stops. The route can be frequented in both directions. The way between two stops is called [i]section[/i] and one of the bus stops is called [i]Zürich[/i]. A bus shall start at Zürich, pass through all the bus stops [b]at least once[/b] and drive along exactly $n+2$ sections before it returns to Zürich in the end. Assuming that the bus can chance directions at each bus stop, how many possible routes are there? EDIT: Sorry, there was a mistake...corrected now, thanks mavropnevma! :oops:

A point $P$ on the side $AB$ of a triangle $ABC$ and points $S$ and $T$ on the sides $AC$ and $BC$ are such that $AP=AS$ and $BP=BT$. The circumcircle of $PST$ meets the sides $AB$ and $BC$ again at $Q$ and $R$, respectively. The lines $PS$ and $QR$ meet at $L$. Prove that the line $CL$ bisects the segment $PQ$. [i]Proposed by A. Antropov[/i]

Prove that in an isosceles triangle $ \triangle ABC$ with $ AC\equal{}BC\equal{}b$ following inequality holds $ b> \pi r$, where $ r$ is inradius.

Three circumferences are tangent to each other, as shown in the figure. The region of the outer circle that is not covered by the two inner circles has an area equal to $2 \pi$. Determine the length of the $PQ$ segment . [img]https://cdn.artofproblemsolving.com/attachments/a/e/65c08c47d4d20a05222a9b6cf65e84a25283b7.png[/img]

Triangle $ABC$ has $AC = 3$, $BC = 5$, $AB = 7$. A circle is drawn internally tangent to the circumcircle of $ABC$ at $C$, and tangent to $AB$. Let $D$ be its point of tangency with $AB$. Find $BD - DA$. [asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(6cm); real labelscalefactor = 2.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.5, xmax = 7.01, ymin = -3, ymax = 8.02; /* image dimensions */ /* draw figures */ draw(circle((1.37,2.54), 5.17)); draw((-2.62,-0.76)--(-3.53,4.2)); draw((-3.53,4.2)--(5.6,-0.44)); draw((5.6,-0.44)--(-2.62,-0.76)); draw(circle((-0.9,0.48), 2.12)); /* dots and labels */ dot((-2.62,-0.76),dotstyle); label("$C$", (-2.46,-0.51), SW * labelscalefactor); dot((-3.53,4.2),dotstyle); label("$A$", (-3.36,4.46), NW * labelscalefactor); dot((5.6,-0.44),dotstyle); label("$B$", (5.77,-0.17), SE * labelscalefactor); dot((0.08,2.37),dotstyle); label("$D$", (0.24,2.61), SW * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); label("$7$",(-3.36,4.46)--(5.77,-0.17), NE * labelscalefactor); label("$3$",(-3.36,4.46)--(-2.46,-0.51),SW * labelscalefactor); label("$5$",(-2.46,-0.51)--(5.77,-0.17), SE * labelscalefactor); /* end of picture */ [/asy]

Find all prime numbers $p$ and $q$ such that $$1 + \frac{p^q - q^p}{p + q}$$ is a prime number. [i]Proposed by Dorlir Ahmeti, Albania[/i]

A median of an acute-angled triangle dissects it into two triangles. Prove that each of them can be covered by a semidisc congruent to a half of the circumdisc of the initial triangle.

In convex quadrilateral $ ABCD$, $ AB \equal{} BC$ and $ AD \equal{} DC$. Point $ E$ lies on segment $ AB$ and point $ F$ lies on segment $ AD$ such that $ B$, $ E$, $ F$, $ D$ lie on a circle. Point $ P$ is such that triangles $ DPE$ and $ ADC$ are similar and the corresponding vertices are in the same orientation (clockwise or counterclockwise). Point $ Q$ is such that triangles $ BQF$ and $ ABC$ are similar and the corresponding vertices are in the same orientation. Prove that points $ A$, $ P$, $ Q$ are collinear.

Let $A$ be a corner of a cube. Let $B$ and $C$ the midpoints of two edges in the positions shown on the figure below: [center][img]http://i.imgur.com/tEODnV0.png[/img][/center] The intersection of the cube and the plane containing $A,B,$ and $C$ is some polygon, $P$. [list=a] [*] How many sides does $P$ have? Justify your answer. [*] Find the ratio of the area of $P$ to the area of $\triangle{ABC}$ and prove that your answer is correct.

A line parallel to the side $AC$ of triangle $ABC$ cuts the side $AB$ at $K$ and the side $BC$ at $M$. $O$ is the intersection point of $AM$ and $CK$. If $AK = AO$ and $KM = MC$, prove that $AM = KB$.

ABC is a triangle with A=90 and C=30.Let M be the midpoint of BC. Let W be a circle passing through A tangent in M to BC. Let P be the circumcircle of ABC. W is intersecting AC in N and P in M. prove that MN is perpendicular to BC.

Let $ABC$ be a triangle, $I$ its incenter and $I_a$ its $A$-excenter. Let $\omega$ be its circuncircle and $D$ be the intersection of $AI$ and $\omega$. Let some line $r$ through $D$ cut $BC$ in $E$ and $\omega$ in $F$. The lines $IE$ and $I_aE$ intersect $I_aF$ and $IF$ in $P$ and $Q$, respectively. Furthermore, the circles $PII_a$ and $QII_a$ intersect $I_aE$ and $IE$ in $R$ and $S$, respectively. Prove that there is a circle passing through $F,E,R$ and $S$.

Consider $5$ points in the plane, such that there are no $3$ of them collinear. Prove that there is a convex quadrilateral with vertices at $4$ points.

Let $ABC$ be a triangle and $D$ be the foot of the altitude from $A$. Let $l$ be the line that passes through the midpoints of $BC$ and $AC$. $E$ is the reflection of $D$ over $l$. Prove that the circumcentre of $\triangle ABC$ lies on the line $AE$.

$ABCD$ is a square with side length 20. A light beam is radiated from $A$ and intersects sides $BC,CD,DA$ respectively and reaches the midpoint of side $AB$. What is the length of the path that the beam has taken? [img]https://s8.uupload.ir/files/photo14908575660_2r3g.jpg[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

Let $AA_1$ and $CC_1$ be altitudes of acute angled triangle $ABC$. A point $D$ is chosen on $AA_1$ such that $A_1D=C_1D$. Let $E$ be the midpoint of $AC$. Prove that points $A$, $C_1$, $D$, $E$ are concylic. [I]Proposed by S. Berlov[/i]

In a triangle $ABC$, the points $A’$, $B’$ and $C’$ on the sides opposite $A$ ,$B$ and $C$, respectively, are such that the lines $AA’$, $BB’$ and $CC’$ are concurrent. Prove that the diameter of the circumscribed circle of the triangle $ABC$ equals the product $|AB’|\cdot |BC’|\cdot |CA’|$ divided by the area of the triangle $A’B’C’$.

Let $ABCD$ be a square of side paper $10$ and $P$ a point on side $BC$. By folding the paper along the $AP$ line, point $B$ determines the point $Q$, as seen in the figure. The line $PQ$ cuts the side $CD$ at $R$. Calculate the perimeter of the triangle $ PCR$ [img]https://3.bp.blogspot.com/-ZSyCUznwutE/XNY7cz7reQI/AAAAAAAAKLc/XqgQnjm8DQYq6Q7fmCAKJwKt3ihoL8AuQCK4BGAYYCw/s400/may%2B2013%2Bl1.png[/img]

Let $PQRS$ be a square with side length 12. Point $A$ lies on segment $\overline{QR}$ with $\angle QPA = 30^\circ$, and point $B$ lies on segment $\overline{PQ}$ with $\angle SRB = 60^\circ$. What is $AB$?

Let $S$ be area of the given parallelogram $ABCD$ and the points $E,F$ belong to $BC$ and $AD$, respectively, such that $BC = 3BE, 3AD = 4AF$. Let $O$ be the intersection of $AE$ and $BF$. Each straightline of $AE$ and $BF$ meets that of $CD$ at points $M$ and $N$, respectively. Determine area of triangle $MON$.

Let $ABCD$ be a tetrahedron with $\angle BAD = 60^{\cdot}$, $\angle BAC = 40^{\cdot}$, $\angle ABD = 80^{\cdot}$, $\angle ABC = 70^{\cdot}$. Prove that the lines $AB$ and $CD$ are perpendicular.

Given triangle $ABC$, squares $ABEF, BCGH, CAIJ$ are constructed externally on side $AB, BC, CA$, respectively. Let $AH \cap BJ = P_1$, $BJ \cap CF = Q_1$, $CF \cap AH = R_1$, $AG \cap CE = P_2$, $BI \cap AG = Q_2$, $CE \cap BI = R_2$. Prove that triangle $P_1 Q_1 R_1$ is congruent to triangle $P_2 Q_2 R_2$.

Let $ABCD$ be an isosceles trapezoid with $AD\parallel BC$ and $BC>AD$ such that the distance between the incenters of $\triangle ABC$ and $\triangle DBC$ is $16$. If the perimeters of $ABCD$ and $ABC$ are $120$ and $114$ respectively, then the area of $ABCD$ can be written as $m\sqrt n,$ where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $100m+n$. [i]Proposed by David Altizio and Evan Chen[/i]