An ancient noble family has $n$ members, each holding a different number of posts . As every year in December, they gather at a very specific place for a Council of War to be held, where also k, from the point of view of the high nobility, unimportant spammers speak up, which, due to their irrelevance, should and cannot be further differentiated. The Council is held as follows: those present speak one after the other, each one carefully put forward his request once. In addition, for reasons of respect, a nobleman never speaks right after a nobleman who holds more posts, while the common people disregarde such rules. Find the number of possible sequences of the Council of war.

Given a circumscribed trapezium $ ABCD$ with circumcircle $ \omega$ and 2 parallel sides $ AD,BC$ ($ BC<AD$). Tangent line of circle $ \omega$ at the point $ C$ meets with the line $ AD$ at point $ P$. $ PE$ is another tangent line of circle $ \omega$ and $ E\in\omega$. The line $ BP$ meets circle $ \omega$ at point $ K$. The line passing through the point $ C$ paralel to $ AB$ intersects with $ AE$ and $ AK$ at points $ N$ and $ M$ respectively. Prove that $ M$ is midpoint of segment $ CN$.

Let $ABCD$ be a convex quadrilateral. Assume line $AB$ and $CD$ intersect at $E$, and $B$ lies between $A$ and $E$. Assume line $AD$ and $BC$ intersect at $F$, and $D$ lies between $A$ and $F$. Assume the circumcircles of $\triangle BEC$ and $\triangle CFD$ intersect at $C$ and $P$. Prove that $\angle BAP=\angle CAD$ if and only if $BD\parallel EF$.

A circle is centered at point $O$ in the plane. Distinct pairs of points $A, B$ and $C, D$ are diametrically opposite on this circle. Point $P$ is chosen on line segment $AD$ such that line $BP$ hits the circle again at $M$ and line $AC$ at $X$ such that $M$ is the midpoint of $PX$. Now, the point $Y \neq X$ is taken for $BX = BY, CD \parallel XY$. IF $\angle PYB = 10^{\circ}$, find the measure of $\angle XCM$. Proposed by Albert Wang (awang11)

A subset $E$ of the set $\{1,2,3,\ldots,50\}$ is said to be [i]special[/i] if it does not contain any pair of the form $\{x,3x\}.$ A special set $E$ is [i]superspecial[/i] if it contains as many elements as possible. How many element there are in a superspecial set and how many superspecial sets there are?

(a) The point $O$ lies inside the convex polygon $A_1A_2A_3...A_n$ . Consider all the angles $A_iOA_j$ where $i, j$ are distinct natural numbers from $1$ to $n$ . Prove that at least $n- 1$ of these angles are not acute . (b) Same problem for a convex polyhedron with $n$ vertices. (V. Boltyanskiy, Moscow)

Let's consider the real numbers $x_1, x_2, x_3, x_4$ satisfying the condition \[ \dfrac{1}{2}\le x_1^2+x_2^2+x_3^2+x_4^2\le 1 \] Find the maximal and the minimal values of expression: \[ A = (x_1 - 2 \cdot x_2 + x_3)^2 + (x_2 - 2 \cdot x_3 + x_4)^2 + (x_2 - 2 \cdot x_1)^2 + (x_3 - 2 \cdot x_4)^2 \]

Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

A $17 \times 17$ square is cut out of a sheet of graph paper. Each cell of this square has one of thenumbers from $1$ to $70$. Prove that there are $4$ distinct squares whose centers $A, B, C, D$ are the vertices of a parallelogramsuch that $AB // CD$, moreover, the sum of the numbers in the squares with centers $A$ and $C$ is equal to that in the squares with centers $B$ and $D$.

An acute triangle $ABC$ ($AB > AC$) has circumcenter $O$, but $D$ is the midpoint of $BC$. Circle with diameter $AD$ intersects sides $AB$ and $AC$ in $E$ and $F$ respectively. On segment $EF$ pick a point $M$ so that $DM \parallel AO$. Prove that triangles $ABD$ and $FDM$ are similar.

The sum of all even positive integers less than $100$ those are not divisible by $3$ is (A): $938$, (B): $940$, (C): $1634$, (D): $1638$, (E): None of the above.

In an acute triangle $ABC$ the altitudes $AA_1$, $BB_1$, $CC_1$ are drawn. A line perpendicular to side $AC$ and passing through a point $A_1$, intersects the line $B_1C_1$ at point $D$. Prove that angle $ADC$ is right.

Let $ \lambda \in (0,2) $ and $ a,b,c,d\in\mathbb{R} $ so that $ a\le b\le c. $ Prove the inequality: $$ (a+b+c+d)^2\ge 4\lambda (ac+bd). $$

Does there exist positive integers $a_{1}<a_{2}<\cdots<a_{100}$ such that for $2 \le k \le 100$, the greatest common divisor of $a_{k-1}$ and $a_{k}$ is greater than the greatest common divisor of $a_{k}$ and $a_{k+1}$?

Bamal, Halvan, and Zuca are playing [i]The Game[/i]. To start, they‘re placed at random distinct vertices on regular hexagon $ABCDEF$. Two or more players collide when they‘re on the same vertex. When this happens, all the colliding players lose and the game ends. Every second, Bamal and Halvan teleport to a random vertex adjacent to their current position (each with probability $\dfrac{1}{2}$), and Zuca teleports to a random vertex adjacent to his current position, or to the vertex directly opposite him (each with probability $\dfrac{1}{3}$). What is the probability that when [i]The Game[/i] ends Zuca hasn‘t lost? [i]Proposed by Edwin Zhao[/i] [hide=Solution][i]Solution.[/i] $\boxed{\dfrac{29}{90}}$ Color the vertices alternating black and white. By a parity argument if someone is on a different color than the other two they will always win. Zuca will be on opposite parity from the others with probability $\dfrac{3}{10}$. They will all be on the same parity with probability $\dfrac{1}{10}$. At this point there are $2 \cdot 2 \cdot 3$ possible moves. $3$ of these will lead to the same arrangement, so we disregard those. The other $9$ moves are all equally likely to end the game. Examining these, we see that Zuca will win in exactly $2$ cases (when Bamal and Halvan collide and Zuca goes to a neighboring vertex). Combining all of this, the answer is $$\dfrac{3}{10}+\dfrac{2}{9} \cdot \dfrac{1}{10}=\boxed{\dfrac{29}{90}}$$ [/hide]

Alice and Bob play a game. First, Alice secretly picks a finite set $S$ of lattice points in the Cartesian plane. Then, for every line $\ell$ in the plane which is horizontal, vertical, or has slope $+1$ or $-1$, she tells Bob the number of points of $S$ that lie on $\ell$. Bob wins if he can determine the set $S$. Prove that if Alice picks $S$ to be of the form \[S = \{(x, y) \in \mathbb{Z}^2 \mid m \le x^2 + y^2 \le n\}\] for some positive integers $m$ and $n$, then Bob can win. (Bob does not know in advance that $S$ is of this form.) [i]Proposed by Mark Sellke[/i]

Given quadratic trinomials $P(x)=x^2+ax+b$ and $Q(x)=x^2+cx+d$, where $a>c$. It is known that for every real $t$ and $s$ with $t+s=1$ the polynomial $B(x)=tP(x)+sQ(x)$ has at least one real root. Prove that $bc \geq ad$.

The average of 10, 4, 8, 7, and 6 is $\textbf{(A)}\ 33 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 7$

Given are two parallel lines $ g$ and $ h$ and a point $ P$, that lies outside of the corridor bounded by $ g$ and $ h$. Construct three lines $ g_1$, $ g_2$ and $ g_3$ through the point $ P$. These lines intersect $ g$ in $ A_1,A_2, A_3$ and $ h$ in $ B_1, B_2, B_3$ respectively. Let $ C_1$ be the intersection of the lines $ A_1B_2$ and $ A_2B_1$, $ C_2$ be the intersection of the lines $ A_1B_3$ and $ A_3B_1$ and let $ C_3$ be the intersection of the lines $ A_2B_3$ and $ A_3B_2$. Show that there exists exactly one line $ n$, that contains the points $ C_1,C_2,C_3$ and that $ n$ is parallel to $ g$ and $ h$.

Let $a_1,b_1,c_1$ be natural numbers. We define \[a_2=\gcd(b_1,c_1),\,\,\,\,\,\,\,\,b_2=\gcd(c_1,a_1),\,\,\,\,\,\,\,\,c_2=\gcd(a_1,b_1),\] and \[a_3=\operatorname{lcm}(b_2,c_2),\,\,\,\,\,\,\,\,b_3=\operatorname{lcm}(c_2,a_2),\,\,\,\,\,\,\,\,c_3=\operatorname{lcm}(a_2,b_2).\] Show that $\gcd(b_3,c_3)=a_2$.

Consider the system of equations $$\log_x y +\log_y z + \log_z x =8$$ $$\log_{\log_y x}z = -3$$ $$\log_z y + \log_x z = 16$$ Find $z.$

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$). Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime. [i]Bobby Shen.[/i]

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Compute $(2 + 0)(2 + 2)(2 + 0)(2 + 2)$. [b]p2.[/b] Given that $25\%$ of $x$ is $120\%$ of $30\%$ of $200$, find $x$. [b]p3.[/b] Jacob had taken a nap. Given that he fell asleep at $4:30$ PM and woke up at $6:23$ PM later that same day, for how many minutes was he asleep? [b]p4.[/b] Kevin is painting a cardboard cube with side length $12$ meters. Given that he needs exactly one can of paint to cover the surface of a rectangular prism that is $2$ meters long, $3$ meters wide, and $6$ meters tall, how many cans of paint does he need to paint the surface of his cube? [b]p5.[/b] How many nonzero digits does $200 \times 25 \times 8 \times 125 \times 3$ have? [b]p6.[/b] Given two real numbers $x$ and $y$, define $x \# y = xy + 7x - y$. Compute the absolute value of $0 \# (1 \# (2 \# (3 \# 4)))$. [b]p7.[/b] A $3$-by-$5$ rectangle is partitioned into several squares of integer side length. What is the fewest number of such squares? Squares in this partition must not overlap and must be contained within the rectangle. [b]p8.[/b] Points $A$ and $B$ lie in the plane so that $AB = 24$. Given that $C$ is the midpoint of $AB$, $D$ is the midpoint of $BC$, $E$ is the midpoint of $AD$, and $F$ is the midpoint of $BD$, find the length of segment $EF$. [b]p9.[/b] Vincent the Bug and Achyuta the Anteater are climbing an infinitely tall vertical bamboo stalk. Achyuta begins at the bottom of the stalk and climbs up at a rate of $5$ inches per second, while Vincent begins somewhere along the length of the stalk and climbs up at a rate of $3$ inches per second. After climbing for $t$ seconds, Achyuta is half as high above the ground as Vincent. Given that Achyuta catches up to Vincent after another $160$ seconds, compute $t$. [b]p10.[/b] What is the minimum possible value of $|x - 2022| + |x - 20|$ over all real numbers $x$? [b]p11.[/b] Let $ABCD$ be a rectangle. Lines $\ell_1$ and $\ell_2$ divide $ABCD$ into four regions such that $\ell_1$ is parallel to $AB$ and line $\ell_2$ is parallel to $AD$. Given that three of the regions have area $6$, $8$, and $12$, compute the sum of all possible areas of the fourth region. [b]p12.[/b] A diverse number is a positive integer that has two or more distinct prime factors. How many diverse numbers are less than $50$? [b]p13.[/b] Let $x$, $y$, and $z$ be real numbers so that $(x+y)(y +z) = 36$ and $(x+z)(x+y) = 4$. Compute $y^2 -x^2$. [b]p14.[/b] What is the remainder when $ 1^{10} + 3^{10} + 7^{10}$ is divided by $58$? [b]p15.[/b] Let $A = (0, 1)$, $B = (3, 5)$, $C = (1, 4)$, and $D = (3, 4)$ be four points in the plane. Find the minimum possible value of $AP + BP + CP + DP$ over all points $P$ in the plane. [b]p16.[/b] In trapezoid $ABCD$, points $E$ and $F$ lie on sides $BC$ and $AD$, respectively, such that $AB \parallel CD \parallel EF$. Given that $AB = 3$, $EF = 5$, and $CD = 6$, the ratio $\frac{[ABEF]}{[CDFE]}$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. (Note: $[F]$ denotes the area of $F$.) [b]p17.[/b] For sets $X$ and $Y$ , let $|X \cap Y |$ denote the number of elements in both $X$ and $Y$ and $|X \cup Y|$ denote the number of elements in at least one of $X$ or $Y$ . How many ordered pairs of subsets $(A,B)$ of $\{1, 2, 3,..., 8\}$ are there such that $|A \cap B| = 2$ and $|A \cup B| = 5$? [b]p18.[/b] A tetromino is a polygon composed of four unit squares connected orthogonally (that is, sharing a edge). A tri-tetromino is a polygon formed by three orthogonally connected tetrominoes. What is the maximum possible perimeter of a tri-tetromino? [b]p19.[/b] The numbers from $1$ through $2022$, inclusive, are written on a whiteboard. Every day, Hermione erases two numbers $a$ and $b$ and replaces them with $ab+a+b$. After some number of days, there is only one number $N$ remaining on the whiteboard. If $N$ has $k$ trailing nines in its decimal representation, what is the maximum possible value of $k$? [b]p20.[/b] Evaluate $5(2^2 + 3^2) + 7(3^2 + 4^2) + 9(4^2 + 5^2) + ... + 199(99^2 + 100^2)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two players are playing a turn based game on a $n \times n$ chessboard. At the beginning, only the bottom left corner of the chessboard contains a piece. At each turn, the player moves the piece to either the square just above, or the square just right, or the diagonal square just right-top. If a player cannot make a move, he loses the game. The game is played once on each $6\times 7$, $6 \times 8$, $7 \times 7$, $7 \times 8$, and $8 \times 8$ chessboard. In how many of them, can the first player guarantee to win? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None} $