Each point with integral coordinates in the plane is coloured white or blue. Prove that one can choose a colour so that for every positive integer $ n $ there exists a triangle of area $ n $ having its vertices of the chosen colour.

Each point in space is colored in one of four different colors. Prove that there is a segment $1$ cm long with endpoints of the same color.

each of the squares in a 2 x 2018 grid of squares is to be coloured black or white such that in any 2 x 2 block , at least one of the 4 squares is white. let P be the number of ways of colouring the grid. find the largest k so that $3^k$ divides P.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Ben was trying to solve for $x$ in the equation $6 + x = 1$. Unfortunately, he was reading upside-down and misread the equation as $1 = x + 9$. What is the positive difference between Ben's answer and the correct answer? [b]p2.[/b] Anjali and Meili each have a chocolate bar shaped like a rectangular box. Meili's bar is four times as long as Anjali's, while Anjali's is three times as wide and twice as thick as Meili's. What is the ratio of the volume of Anjali's chocolate to the volume of Meili's chocolate? [b]p3.[/b] For any two nonnegative integers $m, n$, not both zero, define $m?n = m^n + n^m$. Compute the value of $((2?0)?1)?7$. [b]p4.[/b] Eliza is making an in-scale model of the Phillips Exeter Academy library, and her prototype is a cube with side length $6$ inches. The real library is shaped like a cube with side length $120$ feet, and it contains an entrance chamber in the front. If the chamber in Eliza's model is $0.8$ inches wide, how wide is the real chamber, in feet? [b]p5.[/b] One day, Isaac begins sailing from Marseille to New York City. On the exact same day, Evan begins sailing from New York City to Marseille along the exact same route as Isaac. If Marseille and New York are exactly $3000$ miles apart, and Evan sails exactly 40 miles per day, how many miles must Isaac sail each day to meet Evan's ship in $30$ days? [b]p6.[/b] The conversion from Celsius temperature C to Fahrenheit temperature F is: $$F = 1.8C + 32.$$ If the lowest temperature at Exeter one day was $20^o$ F, and the next day the lowest temperature was $5^o$ C higher, what would be the lowest temperature that day, in degrees Fahrenheit? [b]p7.[/b] In a school, $60\%$ of the students are boys and $40\%$ are girls. Given that $40\%$ of the boys like math and $50\%$ of the people who like math are girls, what percentage of girls like math? [b]p8.[/b] Adam and Victor go to an ice cream shop. There are four sizes available (kiddie, small, medium, large) and seventeen different flavors, including three that contain chocolate. If Victor insists on getting a size at least as large as Adam's, and Adam refuses to eat anything with chocolate, how many different ways are there for the two of them to order ice cream? [b]p9.[/b] There are $10$ (not necessarily distinct) positive integers with arithmetic mean $10$. Determine the maximum possible range of the integers. (The range is defined to be the nonnegative difference between the largest and smallest number within a list of numbers.) [b]p10.[/b] Find the sum of all distinct prime factors of $11! - 10! + 9!$. [b]p11.[/b] Inside regular hexagon $ZUMING$, construct square $FENG$. What fraction of the area of the hexagon is occupied by rectangle $FUME$? [b]p12.[/b] How many ordered pairs $(x, y)$ of nonnegative integers satisfy the equation $4^x \cdot 8^y = 16^{10}$? [b]p13.[/b] In triangle $ABC$ with $BC = 5$, $CA = 13$, and $AB = 12$, Points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, such that $EF \parallel BC$. Given that triangle $AEF$ and trapezoid $EFBC$ have the same perimeter, find the length of $EF$. [b]p14.[/b] Find the number of two-digit positive integers with exactly $6$ positive divisors. (Note that $1$ and $n$ are both counted among the divisors of a number $n$.) [b]p15.[/b] How many ways are there to put two identical red marbles, two identical green marbles, and two identical blue marbles in a row such that no red marble is next to a green marble? [b]p16.[/b] Every day, Yannick submits $8$ more problems to the EMCC problem database than he did the previous day. Every day, Vinjai submits twice as many problems to the EMCC problem database as he did the previous day. If Yannick and Vinjai initially both submit one problem to the database on a Monday, on what day of the week will the total number of Vinjai's problems first exceed the total number of Yannick's problems? [b]p17.[/b] The tiny island nation of Konistan is a cone with height twelve meters and base radius nine meters, with the base of the cone at sea level. If the sea level rises four meters, what is the surface area of Konistan that is still above water, in square meters? [b]p18.[/b] Nicky likes to doodle. On a convex octagon, he starts from a random vertex and doodles a path, which consists of seven line segments between vertices. At each step, he chooses a vertex randomly among all unvisited vertices to visit, such that the path goes through all eight vertices and does not visit the same vertex twice. What is the probability that this path does not cross itself? [b]p19.[/b] In a right-angled trapezoid $ABCD$, $\angle B = \angle C = 90^o$, $AB = 20$, $CD = 17$, and $BC = 37$. A line perpendicular to $DA$ intersects segment $BC$ and $DA$ at $P$ and $Q$ respectively and separates the trapezoid into two quadrilaterals with equal area. Determine the length of $BP$. [b]p20.[/b] A sequence of integers $a_i$ is defined by $a_1 = 1$ and $a_{i+1} = 3i - 2a_i$ for all integers $i \ge 1$. Given that $a_{15} = 5476$, compute the sum $a_1 + a_2 + a_3 + ...+ a_{15}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$A_0A_1\ldots A_n$ is a regular polygon with $n+1$ vertices ($n>2$). Initially $n$ stones are placed at vertex $A_0$. In each allowed operation, $2$ stones are moved simultaneously, at the player's choice: each stone is moved from the vertex where it is located to one of the adjacent $2$ vertices. Find all the values of $n$ for which it is possible to have, after a succession of permitted operations, a stone at each of the vertices $A_1,A_2,\ldots ,A_n$. Clarification: The two stones that move in an allowed operation can be at the same vertex or at different vertices.

For each point $X$ in the plane, a real number $r_X > 0$ is assigned such that $2|r_X - r_Y | \le |XY |$, for any two points $X, Y$ . (Here $|XY |$ denotes the distance between $X$ and $Y$) A frog can jump from $X$ to $Y$ if $r_X = |XY |$. Show that for any two points $X$ and $Y$ , the frog can jump from $X$ to $Y$ in a finite number of steps.

p1. Is there any natural number n such that $n^2 + 5n + 1$ is divisible by $49$ ? Explain. p2. It is known that the parabola $y = ax^2 + bx + c$ passes through the points $(-3,4)$ and $(3,16)$, and does not cut the $x$-axis. Find all possible abscissa values ​​for the vertex point of the parabola. p3. It is known that $T.ABC$ is a regular triangular pyramid with side lengths of $2$ cm. The points $P, Q, R$, and $S$ are the centroids of triangles $ABC$, $TAB$, $TBC$ and $TCA$, respectively . Determine the volume of the triangular pyramid $P.QRS$ . p4. At an event invited $13$ special guests consisting of $ 8$ people men and $5$ women. Especially for all those special guests provided $13$ seats in a special row. If it is not expected two women sitting next to each other, determine the number of sitting positions possible for all those special guests. p5. A table of size $n$ rows and $n$ columns will be filled with numbers $ 1$ or $-1$ so that the product of all the numbers in each row and the product of all the numbers in each column is $-1$. How many different ways to fill the table?

[u]Round 1[/u] [b]p1.[/b] The alphabet of the Aau-Bau language consists of two letters: A and B. Two words have the same meaning if one of them can be constructed from the other by replacing any AA with A, replacing any BB with B, or by replacing any ABA with BAB. For example, the word AABA means the same thing as ABA, and AABA also means the same thing as ABAB. In this language, is it possible to name all seven days of the week? [b]p2.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken. [img]https://cdn.artofproblemsolving.com/attachments/7/6/0fd93a0deaa71a5bb1599d2488f8b4eac5d0eb.jpg[/img] [b]p3.[/b] A playground has a swing-set with exactly three swings. When 3rd and 4th graders from Dr. Anna’s math class play during recess, she has a rule that if a $3^{rd}$ grader is in the middle swing there must be $4^{th}$ graders on that person’s left and right. And if there is a $4^{th}$ grader in the middle, there must be $3^{rd}$ graders on that person’s left and right. Dr. Anna calculates that there are $350$ different ways her students can arrange themselves on the three swings with no empty seats. How many students are in her class? [img]https://cdn.artofproblemsolving.com/attachments/5/9/4c402d143646582376d09ebbe54816b8799311.jpg[/img] [b]p4.[/b] The archipelago Artinagos has $19$ islands. Each island has toll bridges to at least $3$ other islands. An unsuspecting driver used a bad mapping app to plan a route from North Noether Island to South Noether Island, which involved crossing $12$ bridges. Show that there must be a route with fewer bridges. [img]https://cdn.artofproblemsolving.com/attachments/e/3/4eea2c16b201ff2ac732788fe9b78025004853.jpg[/img] [b]p5.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9, ... , 121)$ are in one column? [u]Round 2[/u] [b]p6.[/b] Hungry and Sneaky have opened a rectangular box of chocolates and are going to take turns eating them. The chocolates are arranged in a $2m \times 2n$ grid. Hungry can take any two chocolates that are side-by-side, but Sneaky can take only one at a time. If there are no more chocolates located side-by-side, all remaining chocolates go to Sneaky. Hungry goes first. Each player wants to eat as many chocolates as possible. What is the maximum number of chocolates Sneaky can get, no matter how Hungry picks his? [img]https://cdn.artofproblemsolving.com/attachments/b/4/26d7156ca6248385cb46c6e8054773592b24a3.jpg[/img] [b]p7.[/b] There is a thief hiding in the sultan’s palace. The palace contains $2019$ rooms connected by doors. One can walk from any room to any other room, possibly through other rooms, and there is only one way to do this. That is, one cannot walk in a loop in the palace. To catch the thief, a guard must be in the same room as the thief at the same time. Prove that $11$ guards can always find and catch the thief, no matter how the thief moves around during the search. [img]https://cdn.artofproblemsolving.com/attachments/a/b/9728ac271e84c4954935553c4d58b3ff4b194d.jpg[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are three empty jugs on a table. Winnie the Pooh, Rabbit, and Piglet put walnuts in the jugs one by one. They play successively, with the initial determined by a draw. Thereby Winnie the Pooh plays either in the first or second jug, Rabbit in the second or third, and Piglet in the first or third. The player after whose move there are exactly 1999 walnuts loses the games. Show that Winnie the Pooh and Piglet can cooperate so as to make Rabbit lose.

A sequence $(a_1, a_2,\ldots, a_k)$ consisting of pairwise distinct squares of an $n\times n$ chessboard is called a [i]cycle[/i] if $k\geq 4$ and squares $a_i$ and $a_{i+1}$ have a common side for all $i=1,2,\ldots, k$, where $a_{k+1}=a_1$. Subset $X$ of this chessboard's squares is [i]mischievous[/i] if each cycle on it contains at least one square in $X$. Determine all real numbers $C$ with the following property: for each integer $n\geq 2$, on an $n\times n$ chessboard there exists a mischievous subset consisting of at most $Cn^2$ squares.

In each unit square of an infinite square grid a natural number is written. The polygons of area $n$ with sides going along the gridlines are called [i]admissible[/i], where $n > 2$ is a given natural number. The [i]value [/i] of an admissible polygon is defined as the sum of the numbers inside it. Prove that if the values of any two congruent admissible polygons are equal, then all the numbers written in the unit squares of the grid are equal. (We recall that a symmetric image of polygon $P$ is congruent to $P$.)

Find the number of $(a_1,a_2, ... ,a_{2014})$ permutations of the $(1,2, . . . ,2014)$ such that, for all $1\leq i<j\leq2014$, $i+a_i \leq j+a_j$.

On the grid board shown, a token is placed on each white space. [img]https://cdn.artofproblemsolving.com/attachments/3/2/0060b2436edb0ce25160d2f94f379defef237c.png[/img] A move consists of choosing four squares on the board that form a "$T$" in any of the shapes shown below, and add a token to each of these four squares. [img]https://cdn.artofproblemsolving.com/attachments/8/c/3890aed5289ec9ea2d147f8000a0422c233029.png[/img] Is it possible, after carrying out several moves, to get the $25$ squares to have the same amount of chips?

The numbers $1$ to $n^2$ are written in an n×n squared paper in the usual ordering. Any sequence of right and downwards steps from a square to an adjacent one (by side) starting at square $1$ and ending at square $n^2$ is called a path. Denote by $L(C)$ the sum of the numbers through which path $C$ goes. (a) For a fixed $n$, let $M$ and $m$ be the largest and smallest $L(C)$ possible. Prove that $M-m$ is a perfect cube. (b) Prove that for no $n$ can one find a path $C$ with $L(C ) = 1996$.

In a heap there are $2021$ stones. Two players $A$ and $B$ play removing stones of the pile, alternately starting with $A$. A valid move for $A$ consists of remove $1, 2$ or $7$ stones. A valid move for B is to remove $1, 3, 4$ or $6$ stones. The player who leaves the pile empty after making a valid move wins. Determine if some of the players have a winning strategy. If such a strategy exists, explain it.

In an election, there are a total of $12$ candidates. An election committee has $6$ members voting. It is known that at most two candidates voted by any two committee members are the same. Find the maximum number of committee members.

In line for a SuperRockPop concert were 2005 people. With the aim of offering $3$ tickets for the "backstage", the first person in line was asked to shout "Super", ` the second "Rock", ` the third "Pop", ` the fourth "Super", ` the fifth "Rock", ` the sixth "Pop" and so on. Anyone who said "Rock" or "Pop" was eliminated. This process was repeated, always starting from the first person in the new line, until only $3$ people remained. What positions were these people in at the beginning?

Let positive integer $k$ satisfies $1<k<100$. For the permutation of $1,2,...,100$ be $a_1,a_2,...,a_{100}$, take the minimum $m>k$ such that $a_m$ is at least less than $(k-1)$ numbers of $a_1,a_2,...,a_k$. We know that the number of sequences satisfies $a_m=1$ is $\frac{100!}{4}$. Find the all possible values of $k$.

Given a positive integer $k$, a pigeon and a seagull play a game on an $n\times n$ board. The pigeon goes first, and they take turns doing the operations. The pigeon will choose $m$ grids and lay an egg in each grid he chooses. The seagull will choose a $k\times k$ grids and eat all the eggs inside them. If at any point every grid in the $n\times n $ board has an egg in it, then the pigeon wins. Else, the seagull wins. For every integer $n\geq k$, find all $m$ such that the pigeon wins. [i]Proposed by amano_hina[/i]

For a set $S$ of integers, define $\max (S)$ to be the maximal element of $S$. How many non-empty subsets $S \subseteq \{1, 2, 3, ... , 10\}$ satisfy $\max (S) \le |S| + 2$?

The integers from $1$ to $100$ are written on a “mathlotto” ticket. When you buy a “mathlotto” ticket, you choose $10 $ of these $100$ numbers. Then 10 of the integers from $1$ to $100$ are drawn, and a winning ticket is one which does not contain any of them. Prove that (a) if you buy $13$ tickets, you can choose your numbers so that regardless of which numbers are drawn, you are guaranteed to have at least one winning ticket; (b) if you buy only $12$ tickets, it is possible for you not to have any winning tickets, regardless of how you choose your numbers. (S Tokarev)

Compute the number of functions $f : \{1, 2,... , 2550\} \to \{61, 80, 84\}$ such that $\sum_{k=1}^{2550} f(k)$ is divisible by $3$.

Alex places a rook on any square of an empty $8\times8$ chessboard. Then he places additional rooks one rook at a time, each attacking an odd number of rooks which are already on the board. A rook attacks to the left, to the right, above and below, and only the first rook in each direction. What is the maximum number of rooks Alex can place on the chessboard?

Eight poles stand along the road. A sparrow starts at the first pole and once in a minute flies to a neighboring pole. Let $a(n)$ be the number of ways to reach the last pole in $2n + 1$ flights (we assume $a(m) = 0$ for $m < 3$). Prove that for all $n \ge 4$ $$a(n) - 7a(n-1)+ 15a(n-2) - 10a(n-3) +a(n-4)=0.$$ [i](T. Amdeberhan, F. Petrov )[/i]

All points in a $100 \times 100$ array are colored in one of four colors red, green, blue or yellow in such a way that there are $25$ points of each color in each row and in any column. Prove that there are two rows and two columns such that their four intersection points are all in different colors.