On the ends of the diameter two "$1$"s are written. Each of the semicircles is divided onto two parts and the sum of the numbers of its ends (i.e. "$2$") is written at the midpoint. Then every of the four arcs is halved and in its midpoint the sum of the numbers on its ends is written. Find the total sum of the numbers on the circumference after $n$ steps.

There are $n$ line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his $2n - 1$ friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows: First, he chooses an endpoint of each segment as a “sink”. Then he places the present at the endpoint of the segment he is at. The present moves as follows : $\bullet$ If it is on a line segment, it moves towards the sink. $\bullet$ When it reaches an intersection of two segments, it changes the line segment it travels on and starts moving towards the new sink. If the present reaches an endpoint, the friend on that endpoint can receive their present. Prove that Tony can send presents to exactly $n$ of his $2n - 1$ friends.

We call an even positive integer $n$ [i]nice[/i] if the set $\{1, 2, \dots, n\}$ can be partitioned into $\frac{n}{2}$ two-element subsets, such that the sum of the elements in each subset is a power of $3$. For example, $6$ is nice, because the set $\{1, 2, 3, 4, 5, 6\}$ can be partitioned into subsets $\{1, 2\}$, $\{3, 6\}$, $\{4, 5\}$. Find the number of nice positive integers which are smaller than $3^{2022}$.

Let \( X \) be a finite set of real numbers, \( d \) be a real number, and \(\lambda_1, \lambda_2, \cdots, \lambda_{2025}\) be 2025 non-zero real numbers. Define \[A = \left\{ (x_1, x_2, \cdots, x_{2025}) : x_1, x_2, \cdots, x_{2025} \in X \text{ and } \sum_{i=1}^{2025} \lambda_i x_i = d \right\},\] \[B = \left\{ (x_1, x_2, \cdots, x_{2024}) : x_1, x_2, \cdots, x_{2024} \in X \text{ and } \sum_{i=1}^{2024} (-1)^i x_i = 0 \right\},\] \[C = \left\{ (x_1, x_2, \cdots, x_{2026}) : x_1, x_2, \cdots, x_{2026} \in X \text{ and } \sum_{i=1}^{2026} (-1)^i x_i = 0 \right\}.\] Show that \( |A|^2 \leq |B| \cdot |C| \).

Let $n \ge 3$ be a positive integer. We call a $3 \times 3$ grid [i]beautiful[/i] if the cell located at the center is colored white and all other cells are colored black, or if it is colored black and all other cells are colored white. Determine the minimum value of $a+b$ such that there exist positive integers $a$, $b$ and a coloring of an $a \times b$ grid with black and white, so that it contains $n^2 - n$ [i]beautiful[/i] subgrids.

Graph Airlines $ (GA)$ operates flights between some of the cities of the Republic of Graphia. There are at least three $ GA$ flights from each city, and it is possible to travel from any city in Graphia to any city in Graphia using $ GA$ flights. $ GA$ decides to discontinue some of its flights. Show that this can be done in such a way that it is still possible to travel between any two cities using $ GA$ flights, yet at least $ 2/9$ of the cities have only one flight.

[b]p1.[/b] David runs at $3$ times the speed of Alice. If Alice runs $2$ miles in $30$ minutes, determine how many minutes it takes for David to run a mile. [b]p2.[/b] Al has $2019$ red jelly beans. Bob has $2018$ green jelly beans. Carl has $x$ blue jelly beans. The minimum number of jelly beans that must be drawn in order to guarantee $2$ jelly beans of each color is $4041$. Compute $x$. [b]p3.[/b] Find the $7$-digit palindrome which is divisible by $7$ and whose first three digits are all $2$. [b]p4.[/b] Determine the number of ways to put $5$ indistinguishable balls in $6$ distinguishable boxes. [b]p5.[/b] A certain reduced fraction $\frac{a}{b}$ (with $a,b > 1$) has the property that when $2$ is subtracted from the numerator and added to the denominator, the resulting fraction has $\frac16$ of its original value. Find this fraction. [b]p6.[/b] Find the smallest positive integer $n$ such that $|\tau(n +1)-\tau(n)| = 7$. Here, $\tau(n)$ denotes the number of divisors of $n$. [b]p7.[/b] Let $\vartriangle ABC$ be the triangle such that $AB = 3$, $AC = 6$ and $\angle BAC = 120^o$. Let $D$ be the point on $BC$ such that $AD$ bisect $\angle BAC$. Compute the length of $AD$. [b]p8.[/b] $26$ points are evenly spaced around a circle and are labeled $A$ through $Z$ in alphabetical order. Triangle $\vartriangle LMT$ is drawn. Three more points, each distinct from $L, M$, and $T$ , are chosen to form a second triangle. Compute the probability that the two triangles do not overlap. [b]p9.[/b] Given the three equations $a +b +c = 0$ $a^2 +b^2 +c^2 = 2$ $a^3 +b^3 +c^3 = 19$ find $abc$. [b]p10.[/b] Circle $\omega$ is inscribed in convex quadrilateral $ABCD$ and tangent to $AB$ and $CD$ at $P$ and $Q$, respectively. Given that $AP = 175$, $BP = 147$,$CQ = 75$, and $AB \parallel CD$, find the length of $DQ$. [b]p11. [/b]Let $p$ be a prime and m be a positive integer such that $157p = m^4 +2m^3 +m^2 +3$. Find the ordered pair $(p,m)$. [b]p12.[/b] Find the number of possible functions $f : \{-2,-1, 0, 1, 2\} \to \{-2,-1, 0, 1, 2\}$ that satisfy the following conditions. (1) $f (x) \ne f (y)$ when $x \ne y$ (2) There exists some $x$ such that $f (x)^2 = x^2$ [b]p13.[/b] Let $p$ be a prime number such that there exists positive integer $n$ such that $41pn -42p^2 = n^3$. Find the sum of all possible values of $p$. [b]p14.[/b] An equilateral triangle with side length $ 1$ is rotated $60$ degrees around its center. Compute the area of the region swept out by the interior of the triangle. [b]p15.[/b] Let $\sigma (n)$ denote the number of positive integer divisors of $n$. Find the sum of all $n$ that satisfy the equation $\sigma (n) =\frac{n}{3}$. [b]p16[/b]. Let $C$ be the set of points $\{a,b,c\} \in Z$ for $0 \le a,b,c \le 10$. Alice starts at $(0,0,0)$. Every second she randomly moves to one of the other points in $C$ that is on one of the lines parallel to the $x, y$, and $z$ axes through the point she is currently at, each point with equal probability. Determine the expected number of seconds it will take her to reach $(10,10,10)$. [b]p17.[/b] Find the maximum possible value of $$abc \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^3$$ where $a,b,c$ are real such that $a +b +c = 0$. [b]p18.[/b] Circle $\omega$ with radius $6$ is inscribed within quadrilateral $ABCD$. $\omega$ is tangent to $AB$, $BC$, $CD$, and $DA$ at $E, F, G$, and $H$ respectively. If $AE = 3$, $BF = 4$ and $CG = 5$, find the length of $DH$. [b]p19.[/b] Find the maximum integer $p$ less than $1000$ for which there exists a positive integer $q$ such that the cubic equation $$x^3 - px^2 + q x -(p^2 -4q +4) = 0$$ has three roots which are all positive integers. [b]p20.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle ABC = 60^o$,$\angle ACB = 20^o$. Let $P$ be the point such that $CP$ bisects $\angle ACB$ and $\angle PAC = 30^o$. Find $\angle PBC$. PS. You had better use hide for answers.

Given is a convex octagon $A_1A_2 \ldots A_8$. Given a triangulation $T$, one can take two triangles $\triangle A_iA_jA_k$ and $\triangle A_iA_kA_l$ and replace them with $\triangle A_iA_jA_l$ and $\triangle A_jA_lA_k$. Find the minimal number of operations $k$ we have to do so that for any pair of triangulations $T_1, T_2$, we can reach $T_2$ from $T_1$ using at most $k$ operations.

We have garland with $n$ lights. Some lights are on, some are off. In one move we can take some turned on light (only turned on) and turn off it and also change state of neigbour lights. We want to turn off all lights after some moves.. For what $n$ is it always possible?

Into each box of a $ 2012 \times 2012 $ square grid, a real number greater than or equal to $ 0 $ and less than or equal to $ 1 $ is inserted. Consider splitting the grid into $2$ non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. Suppose that for at least one of the resulting rectangles the sum of the numbers in the boxes within the rectangle is less than or equal to $ 1 $, no matter how the grid is split into $2$ such rectangles. Determine the maximum possible value for the sum of all the $ 2012 \times 2012 $ numbers inserted into the boxes.

In a grid of $100\times 100$ squares, there is a mouse on the top-left square, and there is a piece of cheese in the bottom-right square. The mouse wants to move to the bottom-right square to eat the cheese. For each step, the mouse can move from one square to an adjacent square (two squares are considered adjacent if they share a common edge). Now, any divider can be placed on the common edge of two adjacent squares such that the mouse cannot directly move between these two adjacent squares. A placement of dividers is called "kind" if the mouse can still reach the cheese after the dividers are placed. Find the smallest positive integer $n$ such that, regardless of any "kind" placement of $2023$ dividers, the mouse can reach the cheese in at most $n$ steps.

Given a $2\times 5$ rectangle is divided into unit squares as figure below. [img]https://cdn.artofproblemsolving.com/attachments/6/a/9432bbf40f6d89ee1cbb507e1a3f65326c6a13.png[/img] How many ways are there to write the letters $H, A, N, O, I$ into all of the unit squares, such that two neighbor squares (the squares with a common side) do not contain the same letters? (Each unit square is filled by only one letter and each letter may be used several times or not used as well.)

Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?

[u]Set 2[/u] [b]p4.[/b] $\vartriangle ABC$ is an isosceles triangle with $AB = BC$. Additionally, there is $D$ on $BC$ with $AC = DA = BD = 1$. Find the perimeter of $\vartriangle ABC$. [b]p5[/b]. Let $r$ be the positive solution to the equation $100r^2 + 2r - 1 = 0$. For an appropriate $A$, the infinite series $Ar + Ar^2 + Ar^3 + Ar^4 +...$ has sum $1$. Find $A$. [b]p6.[/b] Let $N(k)$ denote the number of real solutions to the equation $x^4 -x^2 = k$. As $k$ ranges from $-\infty$ to $\infty$, the value of $N(k)$ changes only a finite number of times. Write the sequence of values of $N(k)$ as an ordered tuple (i.e. if $N(k)$ went from $1$ to $3$ to $2$, you would write $(1, 3, 2)$). PS. You should use hide for answers.

Let $ n \geq 3$ and consider a set $ E$ of $ 2n \minus{} 1$ distinct points on a circle. Suppose that exactly $ k$ of these points are to be colored black. Such a coloring is [b]good[/b] if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $ n$ points from $ E$. Find the smallest value of $ k$ so that every such coloring of $ k$ points of $ E$ is good.

Prove that there exists a set $S$ of $3^{1000}$ points in the plane such that for each point $P$ in $S$, there are at least $2000$ points in $S$ whose distance to $P$ is exactly $1$ inch.

Maria and Skyler have a square-shaped cookie with a side length of $1$ inch. They split the cookie by choosing two points on distinct sides of the cookie uniformly at random and cutting across the line segment formed by connecting the two points. If Maria always gets the larger piece, what is the expected amount of extra cookie in Maria’s piece compared to Skyler’s, in square inches?

Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)

Jacob and Laban take turns playing a game. Each of them starts with the list of square numbers $1, 4, 9, \dots, 2021^2$, and there is a whiteboard in front of them with the number $0$ on it. Jacob chooses a number $x^2$ from his list, removes it from his list, and replaces the number $W$ on the whiteboard with $W + x^2$. Laban then does the same with a number from his list, and the repeat back and forth until both of them have no more numbers in their list. Now every time that the number on the whiteboard is divisible by $4$ after a player has taken his turn, Jacob gets a sheep. Jacob wants to have as many sheep as possible. What is the greatest number $K$ such that Jacob can guarantee to get at least $K$ sheep by the end of the game, no matter how Laban plays?

The tourist on an island can play the "getting treasure" game. He has to open a series of doors, each door is colored with one of n colors, according to the following rules: [i](i)[/i] The tourist has n keys, each key with a different color. [i](ii)[/i] Once a key is used, it is not permitted to change until it is destroyed. [i](iii)[/i] Each key can open any door, and keeps intact when it opens the door having different color with it, but is destroyed when it opens the door having the same color with it. Find the least number of doors to ensure that no tourist, no matter how he choose the order of the keys to use, can get the treasure.

We are given a graph $G$ with $n$ edges. For each edge, we write down the lesser degree of two vertices at the end of that edge. Prove that the sum of the resulting $n$ numbers is at most $100n\sqrt{n}$.

A domino is a $1\times2$ or $2\times 1$ rectangle. Diego wants to completely cover a $6\times 6$ board using $18$ dominoes. Determine the smallest positive integer $k$ for which Diego can place $k$ dominoes on the board (without overlapping) such that what remains of the board can be covered uniquely using the remaining dominoes.

The country Dreamland consists of $2016$ cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer $k$ such that no matter how Starways establishes its flights, the cities can always be partitioned into $k$ groups so that from any city it is not possible to reach another city in the same group by using at most $28$ flights. [i]Warut Suksompong, Thailand[/i]

Let $n$ be a positive integer and let $a_n$ be the number of ways to write $n$ as a sum of positive integers, such that any two summands differ by at least $2$. Also, let $b_n$ be the number of ways to write $n$ as a sum of positive integers of the form $5k\pm 1$, $k \in Z$. Prove that $\frac{a_n}{b_n}$ is a constant for all positive integers $n$.

It is known that every student of the class for Sunday once visited the rink, and every boy met there with every girl. Prove that there was a point in time when all the boys, or all the girls of the class were simultaneously on the rink.