A [i]coloring[/i] of the set of integers greater than or equal to $1$, must be done according to the following rule: Each number is colored blue or red, so that the sum of any two numbers (not necessarily different) of the same color is blue. Determine all the possible [i]colorings[/i] of the set of integers greater than or equal to $1$ that follow this rule.

In a $ 2\times 7$ board gridded in $1\times 1$ squares, the $24$ points that are vertices of the squares are considered. [img]https://cdn.artofproblemsolving.com/attachments/9/e/841f11ef9d6fc27cdbe7c91bab6d52d12180e8.gif[/img] Juan and Matías play on this board. Juan paints red the same number of points on each of the three horizontal lines. If Matthias can choose three red dots that are vertices of an acute triangle, Matthias wins the game. What is the maximum number of dots Juan can color in to make sure Matías doesn't win? (For the number found, give an example of coloring that prevents Matías from winning and justify why if the number is greater, Matías can always win.)

Regular $n$-gon is divided to triangles using $n-3$ diagonals of which none of them have common points with another inside polygon. How much among this triangles can there be the most not congruent? [i]Proposed by Dusan Djukic[/i]

Given are the lines $l_1,l_2,\ldots ,l_k$ in the plane, no two of which are parallel and no three of which are concurrent. For which $k$ can one label the intersection points of these lines by $1, 2,\ldots , k-1$ so that in each of the given lines all the labels appear exactly once?

A given finite number of lines in the plane, no two of which are parallel and no three of which are concurrent, divide the plane into finite and infinite regions. In each finite region we write $1$ or $-1$. In one operation, we can choose any triangle made of three of the lines (which may be cut by other lines in the collection) and multiply by $-1$ each of the numbers in the triangle. Determine if it is always possible to obtain $1$ in all the finite regions by successively applying this operation, regardless of the initial distribution of $1$s and $-1$s.

A set of three nonnegative integers $\{x,y,z\}$ with $x < y < z$ is called [i]historic[/i] if $\{z-y,y-x\} = \{1776,2001\}$. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.

The following figure is made up of $12$ segments and $8$ circles. As you can see, at the beginning all the circles are empty. In each operation an empty circle is chosen, it is painted red and inside it the number of red neighboring circles that the chosen circle has is written (in the first operation the chosen circle is painted red and the number $0$ is written). After $8$ operations all the circles are painted red and each one has a number written on it. Prove that, no matter how the operations are done, the sum of all the numbers at the end is the same. [img]https://cdn.artofproblemsolving.com/attachments/3/a/8cd74a0fdc7bb9bc5d1bc764e80ffb58159c0c.png[/img]

Prove that the number $1$ can be written as a sum of $2023$ fractions of the form $\frac{1}{k_i}$, where all nonnegative integers $k_i (1\leq i\leq 2023)$ are distinct.

In the creation of the world there is a lonely island inhabited by dragons, snakes and crocodiles. Every inhabitant eats once a day: every snake eats one dragon for breakfast, every dragon eats one crocodile for lunch and every crocodile eats a snake for dinner. Find the total number of dragons, snakes and crocodiles on the island immediately after the creation of the world (at the beginning of the first day), when, at the end of the sixth day, there is only one inhabitant alive on the island, only one crocodile and during these six days none of the inhabitants of the island considered any to give up their meals due to lack of food.

$n\geq 4$ is an integer number. For any permutation $x_1,x_2,\cdots,x_n$ of the numbers $1,2 \cdots,n$ we write the number $$ x_1+2x_2+\cdots+nx_n $$ on the board. Compute the number of total distinct numbers written on the board.

Andrea and Bruno play a game on a table with $11$ rows and $9$ columns. First Andrea divides the table in $33$ zones. Each zone is formed by $3$ contiguous cells aligned vertically or horizontally, as shown in the figure. [code] ._ |_| |_| _ _ _ |_| |_|_|_| [/code] Then, Bruno writes one of the numbers $0, 1, 2, 3, 4, 5$ in each cell in such a way that the sum of the numbers in each zone is equal to $5$. Bruno wins if the sum of the numbers written in each of the $9$ columns of the table is a prime number. Otherwise, Andrea wins. Show that Bruno always has a winning strategy.

On a $123 \times 123$ board, each square is painted red or blue according to the following conditions: a) Each square painted red that is not on the edge of the board has exactly $5$ blue squares among its $8$ neighboring squares. b) Each square painted blue that is not on the edge of the board has exactly $4$ red squares among its $8$ neighboring squares. Determine the number of red-painted squares on the board.

In a school, more than $90\% $ of the students know both English and German, and more than $90\%$ percent of the students know both English and French. Prove that more than $90\%$ percent of the students who know both German and French also know English.

There are 12 gold stars arranged in a circle on a blue background. Giorgi wants to label each star with one of the letters $G$, $E$, or $O$, such that no two consecutive stars have the same letter. Determine the number of distinct ways Giorgi can label the stars. [img]https://i.imgur.com/qIxdJ8j.jpeg[/img] [i]Proposed by Giorgi Arabidze, Georgia [/i]

How many $5 \times 5$ grids are possible such that each element is either 1 or 0 and each row sum and column sum is $4 ?$ [list=1] [*] 64 [*] 32 [*] 120 [*] 96 [/list]

At a party every girl danced with at least one boy, but not with all of them. Similarly, every boy danced with at least one girl, but not with all of them. Show that there were two girls $G$ and $G'$ and two boys $B$ and $B'$, such that each of $B$ and $G$ danced, $B'$ and $G'$ danced, but $B$ and $G'$ did not dance, and $B'$ and $G$ did not dance.

[b]p1.[/b] Catherine's teacher thinks of a number and asks her to subtract $5$ and then multiply the result by $6$. Catherine accidentally switches the numbers by subtracting 6 and multiplying by $5$ to get $30$. If Catherine had not swapped the numbers, what would the correct answer be? [b]p2.[/b] At Acton Boxborough Regional High School, desks are arranged in a rectangular grid-like configuration. In order to maintain proper social distancing, desks are required to be at least 6 feet away from all other desks. Assuming that the size of the desks is negligible, what is the maximum number of desks that can fit in a $25$ feet by $25$ feet classroom? [b]p3.[/b] Joshua hates writing essays for homework, but his teacher Mr. Meesh assigns two essays every $3$ weeks. However, Mr. Meesh favors Joshua, so he allows Joshua to skip one essay out of every $4$ that are assigned. How many essays does Joshua have to write in a $24$-week school year? [b]p4.[/b] Libra likes to read, but she is easily distracted. If a page number is even, she reads the page twice. If a page number is an odd multiple of three, she skips it. Otherwise, she reads the page exactly once. If Libra's book is $405$ pages long, how many pages in total does she read if she starts on page $1$? (Reading the same page twice counts as two pages.) [b]p5.[/b] Let the GDP of an integer be its Greatest Divisor that is Prime. For example, the GDP of $14$ is $7$. Find the largest integer less than $100$ that has a GDP of $3$. [b]p6.[/b] As has been proven by countless scientific papers, the Earth is a flat circle. Bob stands at a point on the Earth such that if he walks in a straight line, the maximum possible distance he can travel before he falls off is $7$ miles, and the minimum possible distance he can travel before he falls off is $3$ miles. Then the Earth's area in square miles is $k\pi$ for some integer $k$. Compute $k$. [b]p7.[/b] Edward has $2$ magical eggs. Every minute, each magical egg that Edward has will double itself. But there's a catch. At the end of every minute, Edward's brother Eliot will come outside and smash one egg on his forehead, causing Edward to lose that egg permanently. For example, starting with $2$ eggs, after one minute there will be $3$ eggs, then $5$, $9$, and so on. After $1$ hour, the number of eggs can be expressed as $a^b + c$ for positive integers $a$, $b$, $c$ where $a > 1$, and $a$ and $c$ are as small as possible. Find $a + b + c$. [b]p8.[/b] Define a sequence of real numbers $a_1$, $a_2$, $a_3$, $..$, $a_{2019}$, $a_{2020}$ with the property that $a_n =\frac{a_{n-1} + a_n + a_{n+1}}{3}$ for all $n = 2$, $3$, $4$, $5$,$...$, $2018$, $2019$. Given that $a_1 = 1$ and $a_{1000} = 1999$, find $a_{2020}$. [b]p9.[/b] In $\vartriangle ABC$ with $AB = 10$ and $AC = 12$, points $D$ and $E$ lie on sides $\overline{AB}$ and $\overline{AC}$, respectively, such that $AD = 4$ and $AE = 5$. If the area of quadrilateral $BCED$ is $40$, find the area of $\vartriangle ADE$. [b]p10.[/b] A positive integer is called powerful if every prime in its prime factorization is raised to a power greater than or equal to $2$. How many positive integers less than 100 are powerful? [b]p11.[/b] Let integers $A,B < 10, 000$ be the populations of Acton and Boxborough, respectively. When $A$ is divided by $B$, the remainder is $1$. When $B$ is divided by $A$, the remainder is $2020$. If the sum of the digits of $A$ is $17$, find the total combined population of Acton and Boxborough. [b]p12.[/b] Let $a_1$, $a_2$, $...$, $a_n$ be an increasing arithmetic sequence of positive integers. Given $a_n - a_1 = 20$ and $a^2_n - a^2_{n-1} = 63$, find the sum of the terms in the arithmetic sequence. [b]p13.[/b] Bob rolls a cubical, an octahedral and a dodecahedral die ($6$, $8$ and $12$ sides respectively) numbered with the integers from $1$ to $6$, $1$ to $8$ and $1$ to $12$ respectively. If the probability that the sum of the numbers on the cubical and octahedral dice equals the number on the dodecahedral die can be written as $\frac{m}{n}$ , where $m, n$ are relatively prime positive integers, compute $n - m$. [b]p14.[/b] Let $\vartriangle ABC$ be inscribed in a circle with center $O$ with $AB = 13$, $BC = 14$, $AC = 15$. Let the foot of the perpendicular from $A$ to BC be $D$ and let $AO$ intersect $BC$ at $E$. Given the length of $DE$ can be expressed as $\frac{m}{n}$ where $m$, $n$ are relatively prime positive integers, find $m + n$. [b]p15.[/b] The set $S$ consists of the first $10$ positive integers. A collection of $10$ not necessarily distinct integers is chosen from $S$ at random. If a particular number is chosen more than once, all but one of its occurrences are removed. Call the set of remaining numbers $A$. Let $\frac{a}{b}$ be the expected value of the number of the elements in $A$, where $a, b$ are relatively prime positive integers. Find the reminder when $a + b$ is divided by $1000$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.

Given is a positive integer $k$. There are $n$ points chosen on a line, such the distance between any two adjacent points is the same. The points are colored in $k$ colors. For each pair of monochromatic points such that there are no points of the same color between them, we record the distance between these two points. If all distances are distinct, find the largest possible $n$.

In a regular polygon with 100 vertices, 10 vertices are painted blue, and 10 other vertices are painted red. 1. Prove that there exist two distinct blue vertices \( A_1 \) and \( A_2 \), and two distinct red vertices \( R_1 \) and \( R_2 \), such that the distance between \( A_1 \) and \( R_1 \) is equal to the distance between \( A_2 \) and \( R_2 \). 2. Prove that there exist two distinct blue vertices \( A_1 \) and \( A_2 \), and two distinct red vertices \( R_1 \) and \( R_2 \), such that the distance between \( A_1 \) and \( A_2 \) is equal to the distance between \( R_1 \) and \( R_2 \).

The Banana Republic language has more words than letters in its alphabet. Prove that there is a natural number $k$ for which we can choose $k$ different words that use exactly $k$ different letters.

We have a 7x7 board. We want to color some 1x1 squares such that any 3x3 sub-board have more painted 1x1 than no painted 1x1. What is the smallest number of 1x1 that we need to color?

Chip and Dale play on a $100 \times 100$ table. In the beginning, a chess king stands in the upper left corner of the table. At each move the king is moved one square right, down or right-down diagonally. A player cannot move in the direction used by his opponent in the previous move. The players move in turn, Chip begins. The player that cannot move loses. Which player has a winning strategy?

Peter wants to make an unusual die having different positive integers on each of its faces. For neighbouring faces the corresponding numbers should differ by at least two. Find the minimal sum of the six numbers. (Folklore)

The checkered plane (infinitely large house paper) is given. For which pairs (a,, b) one can color each of the squares with one of $a \cdot b$ colors, so that each rectangle of size $ a \times b$ or $b \times a$, placed appropriately in the checkered plane, always contains a unit square with each color ?