On an infinite sheet of graph paper a table is drawn so that in each square of the table stands a number equal to the arithmetic mean of the four adjacent numbers. Out of the table a piece is cut along the lines of the graph paper. Prove that the largest number on the piece always occurs at an edge, where $x = \frac14 (a + b + c + d)$.

Find all possible finite sequences $\{n_0, n_1, n_2, \ldots, n_k \}$ of integers such that for each $i, i$ appears in the sequence $n_i$ times $(0 \leq i \leq k).$

Peter has several $100$ ruble notes and no other money. He starts buying books; each book costs a positive integer number of rubles, and he gets change in $1$ ruble coins. Whenever Peter is buying an expensive book for $100$ rubles or higher he uses only $100$ ruble notes in the minimum necessary number. Whenever he is buying a cheap one (for less than $100$ rubles) he uses his coins if he has enough, otherwise using a $100$ ruble note. When the $100$ ruble notes have come to the end, Peter has expended exactly a half of his money. Is it possible that he has expended $5000$ rubles or more? (Tatiana Kazitsina)

What is the least positive integer $k$ such that, in every convex $101$-gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?

For an integer $n \ge 2$, let $G_n$ be an $n \times n$ grid of unit cells. A subset of cells $H \subseteq G_n$ is considered \textit{quasi-complete} if and only if each row of $G_n$ has at least one cell in $H$ and each column of $G_n$ has at least one cell in $H$. A subset of cells $K \subseteq G_n$ is considered \textit{quasi-perfect} if and only if there is a proper subset $L \subset K$ such that $|L| = n$ and no two elements in $L$ are in the same row or column. Let $\vartheta(n)$ be the smallest positive integer such that every quasi-complete subset $H \subseteq G_n$ with $|H| \ge \vartheta(n)$ is also quasi-perfect. Moreover, let $\varrho(n)$ be the number of quasi-complete subsets $H \subseteq G_n$ with $|H| = \vartheta(n) - 1$ such that $H$ is not quasi-perfect. Compute $\vartheta(20) + \varrho(20)$.

Metadieu, Tercieu and Quartieu are three bodybuilder warriors who fight against an $n$-headed monster. Each of them can attack the monster according to the following rules: (1) Metadieu's attack consists of cutting off half of the monster's heads, then cutting off one more head. If the monster's number of heads is odd, Metadieu cannot attack; (2) Tercieu's attack consists of cutting off a third of the monster's heads, then cutting off two more heads. If the monster's number of heads is not a multiple of 3, Tercieu cannot attack; (3) Quartieu's attack consists of cutting off a quarter of the monster's heads, then cutting off three more heads. If the monster's number of heads is not a multiple of 4, Quartieu cannot attack; If none of the three warriors can attack the monster at some point, then it will devour our three heroes. The objective of the three warriors is to defeat the monster, and for that they need to cut off all its heads, one warrior attacking at a time. For what positive integer values of $n$ is it possible for the three warriors to combine a sequence of attacks in order to defeat the monster?

A positive integer is written on the board. Every minute Maxim adds to the number on the board one of its positive divisors, writes the result on the board and erases the previous number. However, it is forbidden for him to add the same number twice in a row. Prove that he can proceed in such a way that eventually a perfect square will appear on the board.

The closed line segment $I$ is covered by finitely many closed line segments. Show that one can choose a subfamily $S$ of the family of line segments having the properties: (1) the chosen line segments are disjoint, (2) the sum of the lengths of the line segments of S is more than half of the length of $I.$ Show that the claim does not hold any more if the line segment $I$ is replaced by a circle and other occurences of the compound word ''line segment" by the word ''circular arc".

There are 2014 houses in a circle. Let $A$ be one of these houses. Santa Claus enters house $A$ and leaves a gift. Then with probability $1/2$ he visits $A$'s left neighbor and with probability $1/2$ he visits $A$'s right neighbor. He leaves a gift also in that second house, and then repeats the procedure (visits with probability $1/2$ either of the neighbors, leaves a gift, etc). Santa finishes as soon as every house has received at least one gift. Prove that any house $B$ different from $A$ has a probability of $1/2013$ of being the last house receiving a gift.

In the country some mathematicians know each other and any division of them into two sets contain 2 friends from different sets.It is known that if you put any set of four or more mathematicians at a round table so that any two neighbours know each other , then at the table there are two friends not sitting next to each other.We denote by $c_i $ the number of sets of $i$ pairwise familiar mathematicians(by saying "familiar" it means know each other).Prove that $c_1-c_2+c_3-c_4+...=1$

Each of the numbers in the set $N = \{1, 2, 3, \cdots, n - 1\}$, where $n \geq 3$, is colored with one of two colors, say red or black, so that: [i](i)[/i] $i$ and $n - i$ always receive the same color, and [i](ii)[/i] for some $j \in N$, relatively prime to $n$, $i$ and $|j - i|$ receive the same color for all $i \in N, i \neq j.$ Prove that all numbers in $N$ must receive the same color.

Consider $n$ different points lying on a circle, such that there are not three chords defined by that point that pass through the same interior point of the circle. a) Find the value of $n$, if the numbers of triangles that are defined using $3$ of the n points is equal to $2n$ b) Find the value of $n$, if the numbers of the intersection points of the chords that are interior to the circle is equal to $5n$.

A convex pentagon $ABCDE$ satisfies that the sidelengths and $AC,AD\leq \sqrt 3.$ Let us choose $2011$ distinct points inside this pentagon. Prove that there exists an unit circle with centre on one edge of the pentagon, and which contains at least $403$ points out of the $2011$ given points. {Edited} {I posted it correctly before but because of a little confusion deleted the sidelength part, sorry.}

There are $2022$ students in winter school. Two arbitrary students are friend or enemy each other. Each turn, we choose a student $S$, make friends of $S$ enemies, and make enemies of $S$ friends. This continues until it satisfies the final condition. [b]Final Condition[/b] : For any partition of students into two non-empty groups $A$, $B$, there exist two students $a$, $b$ such that $a\in A$, $b\in B$, and $a$, $b$ are friend each other. Determine the minimum value of $n$ such that regardless of the initial condition, we can satisfy the final condition with no more than $n$ turns.

Let $ n $ and $ r $ be positive integers and $ A $ be a set of lattice points in the plane such that any open disc of radius $ r $ contains a point of $ A $. Show that for any coloring of the points of $ A $ in $ n $ colors there exists four points of the same color which are the vertices of a rectangle.

$1000$ children, no two of the same height, lined up. Let us call a pair of different children $(a,b)$ good if between them there is no child whose height is greater than the height of one of $a$ and $b$, but less than the height of the other. What is the greatest number of good pairs that could be formed? (Here, $(a,b)$ and $(b,a)$ are considered the same pair.) [i]Proposed by I. Bogdanov[/i]

In a country there are $2021$ cities. Each pair of cities is either linked by a single road or not linked at all. It is known that for any subset of $2019$ cities, the total number of roads between them is the same. If the total number of roads in that country is $A$, find all possible values of $A$.

For a finite set $A$ of positive integers and its subset $B$, call $B$ a [i]half subset[/i] of $A$ when it satisfies the equation $\sum_{a\in A}a=2\sum_{b\in B}b$. For example, if $A=\{1,2,3\}$, then $\{1,2\}$ and $\{3\}$ are half subset of $A$. Determine all positive integers $n$ such that there exists a finite set $A$ which has exactly $n$ half subsets.

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

A given equilateral triangle of side $10$ is divided into $100$ equilateral triangles of side $1$ by drawing parallel lines to the sides of the original triangle. Find the number of equilateral triangles, having vertices in the intersection points of parallel lines whose sides lie on the parallel lines.

Let $ x_1$, $ x_2$, $ \ldots$, $ x_n$ be real numbers satisfying the conditions: \[ \left\{\begin{array}{cccc} |x_1 \plus{} x_2 \plus{} \cdots \plus{} x_n | & \equal{} & 1 & \ \\ |x_i| & \leq & \displaystyle \frac {n \plus{} 1}{2} & \ \textrm{ for }i \equal{} 1, 2, \ldots , n. \end{array} \right. \] Show that there exists a permutation $ y_1$, $ y_2$, $ \ldots$, $ y_n$ of $ x_1$, $ x_2$, $ \ldots$, $ x_n$ such that \[ | y_1 \plus{} 2 y_2 \plus{} \cdots \plus{} n y_n | \leq \frac {n \plus{} 1}{2}. \]

In a certain group there are $n \ge 5$ people, with every two people who do not know each other exactly having one mutual friend and no one knows everyone else. Prove $5$ of $n$ people, may sit at a circle around the table so that each of them sits between a) friends, b) strangers.

[b]p1.[/b] Roll two dice. What is the probability that the sum of the rolls is prime? [b]p2. [/b]Compute the sum of the first $20$ squares. [b]p3.[/b] How many integers between $0$ and $999$ are not divisible by $7, 11$, or $13$? [b]p4.[/b] Compute the number of ways to make $50$ cents using only pennies, nickels, dimes, and quarters. [b]p5.[/b] A rectangular prism has side lengths $1, 1$, and $2$. What is the product of the lengths of all of the diagonals? [b]p6.[/b] What is the last digit of $7^{6^{5^{4^{3^{2^1}}}}}$ ? [b]p7.[/b] Given square $ABCD$ with side length $3$, we construct two regular hexagons on sides $AB$ and $CD$ such that the hexagons contain the square. What is the area of the intersection of the two hexagons? [img]https://cdn.artofproblemsolving.com/attachments/f/c/b2b010cdd0a270bc10c6e3bb3f450ba20a03e7.png[/img] [b]p8.[/b] Brooke is driving a car at a steady speed. When she passes a stopped police officer, she begins decelerating at a rate of $10$ miles per hour per minute until she reaches the speed limit of $25$ miles per hour. However, when Brooke passed the police officer, he immediately began accelerating at a rate of $20$ miles per hour per minute until he reaches the rate of $40$ miles per hour. If the police officer catches up to Brooke after 3 minutes, how fast was Brooke driving initially? [b]p9.[/b] Find the ordered pair of positive integers $(x, y)$ such that $144x - 89y = 1$ and $x$ is minimal. [b]p10.[/b] How many zeroes does the product of the positive factors of $10000$ (including $1$ and $10000$) have? [b]p11.[/b] There is a square configuration of desks. It is known that one can rearrange these desks such that it has $7$ fewer rows but $10$ more columns, with $13$ desks remaining. How many desks are there in the square configuration? [b]p12.[/b] Given that there are $168$ primes with $3$ digits or less, how many numbers between $1$ and $1000$ inclusive have a prime number of factors? [b]p13.[/b] In the diagram below, we can place the integers from $1$ to $19$ exactly once such that the sum of the entries in each row, in any direction and of any size, is the same. This is called the magic sum. It is known that such a configuration exists. Compute the magic sum. [img]https://cdn.artofproblemsolving.com/attachments/3/4/7efaa5ba5ad250e24e5ad7ef03addbf76bcfb4.png[/img] [b]p14.[/b] Let $E$ be a random point inside rectangle $ABCD$ with side lengths $AB = 2$ and $BC = 1$. What is the probability that angles $ABE$ and $CDE$ are both obtuse? [b]p15.[/b] Draw all of the diagonals of a regular $13$-gon. Given that no three diagonals meet at points other than the vertices of the $13$-gon, how many intersection points lie strictly inside the $13$-gon? [b]p16.[/b] A box of pencils costs the same as $11$ erasers and $7$ pencils. A box of erasers costs the same as $6$ erasers and a pencil. A box of empty boxes and an eraser costs the same as a pencil. Given that boxes cost a penny and each of the boxes contain an equal number of objects, how much does it costs to buy a box of pencils and a box of erasers combined? [b]p17.[/b] In the following figure, all angles are right angles and all sides have length $1$. Determine the area of the region in the same plane that is at most a distance of $1/2$ away from the perimeter of the figure. [img]https://cdn.artofproblemsolving.com/attachments/6/2/f53ae3b802618703f04f41546e3990a7d0640e.png[/img] [b]p18.[/b] Given that $468751 = 5^8 + 5^7 + 1$ is a product of two primes, find both of them. [b]p19.[/b] Your wardrobe contains two red socks, two green socks, two blue socks, and two yellow socks. It is currently dark right now, but you decide to pair up the socks randomly. What is the probability that none of the pairs are of the same color? [b]p20.[/b] Consider a cylinder with height $20$ and radius $14$. Inside the cylinder, we construct two right cones also with height $20$ and radius $14$, such that the two cones share the two bases of the cylinder respectively. What is the volume ratio of the intersection of the two cones and the union of the two cones? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A chess tournament took place between $2n+1$ players. Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly $k$ games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than $n-\sqrt{2k}$ and no more than $n+\sqrt{2k}$ games.

On a circumference at some points sit $12$ grasshoppers. The points divide the circumference into $12$ arcs. By a signal each grasshopper jumps from its point to the midpoint of its arc (in clockwise direction). In such way new arcs are created. The process repeats for a number of times. Can it happen that at least one of the grasshoppers returns to its initial point after a) $12$ jumps? (4) a) $13$ jumps? (3)