In an inscribed quadrilateral $ABCD$ we have $BC = CD$. Prove that the area of the quadrilateral is equal to $\frac{(AC)^2 \sin A}{2}$ (D. Fomin, Leningrad)

Let $ABC$ be a scalene triangle. Points $A_1,B_1$ and $C_1$ are chosen on segments $BC,CA$ and $AB$, respectively, such that $\triangle A_1B_1C_1$ and $\triangle ABC$ are similar. Let $A_2$ be the unique point on line $B_1C_1$ such that $AA_2=A_1A_2$. Points $B_2$ and $C_2$ are defined similarly. Prove that $\triangle A_2B_2C_2$ and $\triangle ABC$ are similar. [i]Fedir Yudin [/i]

Triangle $ABC$ is inscribed in circle $w$. Line $l_{1}$ bisects $\angle BAC$ and meets segments $BC$ and $w$ in $D$ and $M$,respectively. Let $y$ denote the circle centered at $M$ with radius $BM$. Line $l_{2}$ passes through $D$ and meets circle $y$ at $X$ and $Y$. Prove that line $l_{1}$ also bisects $\angle XAY$

Let $ABCD$ a convex quadrilateral (this means that all interior angles are smaller than $180^o$), such that there exist a point $M$ on line segment $AB$ and a point $N$ on line segment $BC$ having the property that $AN$ cuts the quadrilateral in two parts of equal area, and such that the same property holds for $CM$. Prove that $MN$ cuts the diagonal $BD$ in two segments of equal length.

In $\triangle ABC$, $\left|\overline{AB}\right|=\left|\overline{AC}\right|$, $D$ is the foot of the perpendicular from $C$ to $AB$ and $E$ the foot of the perpendicular from $B$ to $AC$, then $\textbf{(A)}~\left|\overline{BC}\right|^3>\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(B)}~\left|\overline{BC}\right|^3<\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(C)}~\left|\overline{BC}\right|^3=\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(D)}~\text{None of the above}$

Let $A$, $B$, $C$, $D$ be four points in the plane, with $C$ and $D$ on the same side of the line $AB$, such that $AC \cdot BD = AD \cdot BC$ and $\angle ADB = 90^{\circ}+\angle ACB$. Find the ratio \[\frac{AB \cdot CD}{AC \cdot BD}, \] and prove that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles $ACD$ and $BCD$ at the point $C$ are perpendicular.)

Let $\Omega$ be the circumcircle of cyclic quadrilateral $ABCD$. Consider such pairs of points $P$, $Q$ of diagonal $AC$ that the rays $BP$ and $BQ$ are symmetric with respect the bisector of angle $B$. Find the locus of circumcenters of triangles $PDQ$.

A cube is constructed from $4$ white unit cubes and $4$ black unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

A triangle is cut by $3$ cevians from its $3$ vertices into $7$ pieces: $4$ triangles and $3$ quadrilaterals. Determine if it is possible that all $3$ quadrilaterals are inscribed.

Triangle $ABC$ is inscribed in a circle with center $O$. Let $q$ be the circle passing through $A$, $O$ and $B$. The lines $CA$ and $CB$ intersect $q$ at the points $D$ and $E$ (different from $A$ and $B$). Prove that the lines $CO$ and $DE$ are perpendicular to each other. (S Markelov)

Two disjoint circles $\omega_1$ and $\omega_2$ are inscribed into an angle. Consider all pairs of parallel lines $l_1$ and $l_2$ such that $l_1$ touches $\omega_1$ and $l_2$ touches $\omega_2$ ($\omega_1$, $\omega_2$ lie between $l_1$ and $l_2$). Prove that the medial lines of all trapezoids formed by $l_1$ and $l_2$ and the sides of the angle touch some fixed circle.

The area of a convex polygon in the plane is equally shared by the four standard quadrants, and all non-zero lattice points lie outside the polygon. Show that the area of the polygon is less than $4$.

Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.

Let $n$ be a positive integer and $m = \frac{(n+1)(n+2)}{2}$. In coordinate plane, there are $n$ distinct lines $L_1, L_2, \ldots, L_n$ and $m$ distinct points $A_1, A_2, \ldots, A_m$, satisfying the following conditions: [b][i]i)[/i][/b] Any two lines are non-parallel. [b][i]ii)[/i][/b] Any three lines are non-concurrent. [b][i]iii)[/i][/b] Only $A_1$ does not lies on any line $L_k$, and there are exactly $k + 1$ points $A_j$'s that lie on line $L_k$ $(k = 1, 2, \ldots, n).$ Prove that there exist a unique polynomial $p(x, y)$ with degree $n$ satisfying $p(A_1) = 1$ and $p(A_j) = 0$ for $j = 2, 3, \ldots, m.$

Right triangle $ABC$ has hypotenuse $AB = 26$, and the inscribed circle of $ABC$ has radius $5$. The largest possible value of $BC$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are both positive integers. Find $100m + n$. [i]Proposed by Jason Xia[/i]

[u]Round 1[/u] [b]p1.[/b] Jom and Terry both flip a fair coin. What is the probability both coins show the same side? [b]p2.[/b] Under the same standard air pressure, when measured in Fahrenheit, water boils at $212^o$ F and freezes at $32^o$ F. At thesame standard air pressure, when measured in Delisle, water boils at $0$ D and freezes at $150$ D. If x is today’s temperature in Fahrenheit and y is today’s temperature expressed in Delisle, we have $y = ax + b$. What is the value of $a + b$? (Ignore units.) [b]p3.[/b] What are the last two digits of $5^1 + 5^2 + 5^3 + · · · + 5^{10} + 5^{11}$? [u]Round 2[/u] [b]p4.[/b] Compute the average of the magnitudes of the solutions to the equation $2x^4 + 6x^3 + 18x^2 + 54x + 162 = 0$. [b]p5.[/b] How many integers between $1$ and $1000000$ inclusive are both squares and cubes? [b]p6.[/b] Simon has a deck of $48$ cards. There are $12$ cards of each of the following $4$ suits: fire, water, earth, and air. Simon randomly selects one card from the deck, looks at the card, returns the selected card to the deck, and shuffles the deck. He repeats the process until he selects an air card. What is the probability that the process ends without Simon selecting a fire or a water card? [u]Round 3[/u] [b]p7.[/b] Ally, Beth, and Christine are playing soccer, and Ally has the ball. Each player has a decision: to pass the ball to a teammate or to shoot it. When a player has the ball, they have a probability $p$ of shooting, and $1 - p$ of passing the ball. If they pass the ball, it will go to one of the other two teammates with equal probability. Throughout the game, $p$ is constant. Once the ball has been shot, the game is over. What is the maximum value of $p$ that makes Christine’s total probability of shooting the ball $\frac{3}{20}$ ? [b]p8.[/b] If $x$ and $y$ are real numbers, then what is the minimum possible value of the expression $3x^2 - 12xy + 14y^2$ given that $x - y = 3$? [b]p9.[/b] Let $ABC$ be an equilateral triangle, let $D$ be the reflection of the incenter of triangle $ABC$ over segment $AB$, and let $E$ be the reflection of the incenter of triangle $ABD$ over segment $AD$. Suppose the circumcircle $\Omega$ of triangle $ADE$ intersects segment $AB$ again at $X$. If the length of $AB$ is $1$, find the length of $AX$. [u]Round 4[/u] [b]p10.[/b] Elaine has $c$ cats. If she divides $c$ by $5$, she has a remainder of $3$. If she divides $c$ by $7$, she has a remainder of $5$. If she divides $c$ by $9$, she has a remainder of $7$. What is the minimum value $c$ can be? [b]p11.[/b] Your friend Donny offers to play one of the following games with you. In the first game, he flips a fair coin and if it is heads, then you win. In the second game, he rolls a $10$-sided die (its faces are numbered from $1$ to $10$) $x$ times. If, within those $x$ rolls, the number $10$ appears, then you win. Assuming that you like winning, what is the highest value of $x$ where you would prefer to play the coin-flipping game over the die-rolling game? [b]p12.[/b] Let be the set $X = \{0, 1, 2, ..., 100\}$. A subset of $X$, called $N$, is defined as the set that contains every element $x$ of $X$ such that for any positive integer $n$, there exists a positive integer $k$ such that n can be expressed in the form $n = x^{a_1}+x^{a_2}+...+x^{a_k}$ , for some integers $a_1, a_2, ..., a_k$ that satisfy $0 \le a_1 \le a_2 \le ... \le a_k$. What is the sum of the elements in $N$? PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2782880p24446580]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A line $r$ contains the points $A$, $B$, $C$, $D$ in that order. Let $P$ be a point not in $r$ such that $\angle{APB} = \angle{CPD}$. Prove that the angle bisector of $\angle{APD}$ intersects the line $r$ at a point $G$ such that: $\frac{1}{GA} + \frac{1}{GC} = \frac{1}{GB} + \frac{1}{GD}$

Let $ABC$ be an isosceles triangle with $\angle A=\angle B=80^\circ$. A straight line passes through $B$ and through the circumcenter of the triangle and intersects the side $AC$ at $D$. Prove that $AB=CD$.

Let $ABC$ be a triangle. Point $E,F$ moves on the opposite ray of $BA,CA$ such that $BF=CE$. Let $M,N$ be the midpoint of $BE,CF$. $BF$ cuts $CE$ at $D$ a) Suppost that $I$ is the circumcenter of $(DBE)$ and $J$ is the circumcenter of $(DCF)$, Prove that $MN \parallel IJ$ b) Let $K$ be the midpoint of $MN$ and $H$ be the orthocenter of triangle $AEF$. Prove that when $E$ varies on the opposite ray of $BA$, $HK$ go through a fixed point

Let $AB$ and $FD$ be chords in circle, which does not intersect and $P$ point on arc $AB$ which does not contain chord $FD$. Lines $PF$ and $PD$ intersect chord $AB$ in $Q$ and $R$. Prove that $\frac{AQ* RB}{QR}$ is constant, while point $P$ moves along the ray $AB$.

[u]Round 1[/u] [b]p1.[/b] Six pirates – Captain Jack and his five crewmen – sit in a circle to split a treasure of $99$ gold coins. Jack must decide how many coins to take for himself and how many to give each crewman (not necessarily the same number to each). The five crewmen will then vote on Jack's decision. Each is greedy and will vote “aye” only if he gets more coins than each of his two neighbors. If a majority vote “aye”, Jack's decision is accepted. Otherwise Jack is thrown overboard and gets nothing. What is the most coins Captain Jack can take for himself and survive? [b]p2[/b]. Rose and Bella take turns painting cells red and blue on an infinite piece of graph paper. On Rose's turn, she picks any blank cell and paints it red. Bella, on her turn, picks any blank cell and paints it blue. Bella wins if the paper has four blue cells arranged as corners of a square of any size with sides parallel to the grid lines. Rose goes first. Show that she cannot prevent Bella from winning. [img]https://cdn.artofproblemsolving.com/attachments/d/6/722eaebed21a01fe43bdd0dedd56ab3faef1b5.png[/img] [b]p3.[/b] A $25\times 25$ checkerboard is cut along the gridlines into some number of smaller square boards. Show that the total length of the cuts is divisible by $4$. For example, the cuts shown on the picture have total length $16$, which is divisible by $4$. [img]https://cdn.artofproblemsolving.com/attachments/c/1/e152130e48b804fe9db807ef4f5cd2cbad4947.png[/img] [b]p4.[/b] Each robot in the Martian Army is equipped with a battery that lasts some number of hours. For any two robots, one's battery lasts at least three times as long as the other's. A robot works until its battery is depleted, then recharges its battery until it is full, then goes back to work, and so on. A battery that lasts $N$ hours takes exactly $N$ hours to recharge. Prove that there will be a moment in time when all the robots are recharging (so you can invade the planet). [b]p5.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png [/img] [u]Round 2[/u] [b]p6.[/b] The sum of $2015$ rational numbers is an integer. The product of every pair of them is also an integer. Prove that they are all integers. (A rational number is one that can be written as $m/n$, where $m$ and $n$ are integers and $n\ne 0$.) [b]p7.[/b] An $N \times N$ table is filled with integers such that numbers in cells that share a side differ by at most $1$. Prove that there is some number that appears in the table at least $N$ times. For example, in the $5 \times 5$ table below the numbers $1$ and $2$ appear at least $5$ times. [img]https://cdn.artofproblemsolving.com/attachments/3/8/fda513bcfbe6834d88fb8ca0bfcdb504d8b859.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

[u]Round 1[/u] [b]p1.[/b] Let $ABCDEF$ be a regular hexagon. How many acute triangles have all their vertices among the vertices of $ABCDEF$? [b]p2.[/b] A rectangle has a diagonal of length $20$. If the width of the rectangle is doubled, the length of the diagonal becomes $22$. Given that the width of the original rectangle is $w$, compute $w^2$. [b]p3.[/b] The number $\overline{2022A20B22}$ is divisible by 99. What is $A + B$? [u]Round 2[/u] [b]p4.[/b] How many two-digit positive integers have digits that sum to at least $16$? [b]p5.[/b] For how many integers $k$ less than $10$ do there exist positive integers x and y such that $k =x^2 - xy + y^2$? [b]p6.[/b] Isosceles trapezoid $ABCD$ is inscribed in a circle of radius $2$ with $AB \parallel CD$, $AB = 2$, and one of the interior angles of the trapezoid equal to $110^o$. What is the degree measure of minor arc $CD$? [u]Round 3[/u] [b]p7.[/b] In rectangle $ALEX$, point $U$ lies on side $EX$ so that $\angle AUL = 90^o$. Suppose that $UE = 2$ and $UX = 12$. Compute the square of the area of $ALEX$. [b]p8.[/b] How many digits does $20^{22}$ have? [b]p9.[/b] Compute the units digit of $3 + 3^3 + 3^{3^3} + ... + 3^{3^{...{^3}}}$ , where the last term of the series has $2022$ $3$s. [u]Round 4[/u] [b]p10.[/b] Given that $\sqrt{x - 1} + \sqrt{x} = \sqrt{x + 1}$ for some real number $x$, the number $x^2$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p11.[/b] Eric the Chicken Farmer arranges his $9$ chickens in a $3$-by-$3$ grid, with each chicken being exactly one meter away from its closest neighbors. At the sound of a whistle, each chicken simultaneously chooses one of its closest neighbors at random and moves $\frac12$ of a unit towards it. Given that the expected number of pairs of chickens that meet can be written as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers, compute $p + q$. [b]p12.[/b] For a positive integer $n$, let $s(n)$ denote the sum of the digits of $n$ in base $10$. Find the greatest positive integer $n$ less than $2022$ such that $s(n) = s(n^2)$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2949432p26408285]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\Theta_1$ and $\Theta_2$ be circumferences with centers $O_1$ and $O_2$, exteriorly tangents. Let $A$ and $B$ be points in $\Theta_1$ and $\Theta_2$, respectively, such that $AB$ is common external tangent to $\Theta_1$ and $\Theta_2$. Let $C$ and $D$ be points on the semiplane determined by $AB$ that does not contain $O_1$ and $O_2$ such that $ABCD$ is a square. If $O$ is the center of this square, compute the possible values for the angle $\angle O_1OO_2$.

The lines with equations $ax-2y=c$ and $2x+by=-c$ are perpendicular and intersect at $(1, -5)$. What is $c$? $\textbf{(A) } -13\qquad \textbf{(B) } -8\qquad \textbf{(C) } 2\qquad \textbf{(D) } 8\qquad \textbf{(E) } 13$