At a chess tournament, every pair of contestants played each other at most once. If any two con- testants, $A$ and $B$, failed to play each other, then exactly two other contestants, $C$ and $D$, played against both $A$ and $B$ during the tournament. Moreover, no $4$ contestants played exactly $5$ games between them. Prove that every contestant played the same number of games. [i]Authored by Mirko Petrushevski[/i]

Each face of a cube is painted with a different color. How many distinct cubes can be created this way? (*Observation: The ways to color the cube are $6!$, since each time a color is used on one face, there is one fewer available for the others. However, this does not determine $6!$ different cubes, since colorings that differ only by rotation should be considered the same.*)

Let $ n(\ge2) $ be a positive integer. Find the minimum $ m $, so that there exists $x_{ij}(1\le i ,j\le n)$ satisfying: (1)For every $1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\} $ or $ x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}.$ (2)For every $1\le i \le n$, there are at most $m$ indices $k$ with $x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}.$ (3)For every $1\le j \le n$, there are at most $m$ indices $k$ with $x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.$

For a randomly selected permutation $ \mathbf{f} = (f_1,..., f_n) $ of the set $ \{1,\ldots, n\} $ let us denote by $ X(\mathbf{f}) $ the largest number $ k \leq n $ such that $ f_i < f_{ i+1} $ for all numbers $ i < k $. Prove that the expected value of the random variable $ X $ is $ \sum_{k=1}^n \frac{1}{k!} $.

We call a graph symmetric, if we can put its vertices on the plane such that if the edges are segments, the graph has a reflectional symmetry with respect to a line not passing through its vertices. Find the least value of $K$ such that the edges of every graph with $100$ vertices, can be divided into $K$ symmetric subgraphs.

In a town there are four pubs, $A,B,C,D$, and any two of them are connected to each other except $A$ and $D$. A drunkard wanders about the pubs starting with $A$ and, after having a drink, goes to any of the pubs directly connected, with equal probability. (a) What is the probability that the drunkard is at $C$ at its fifth drink? (b) Where is the drunkard most likely to be after $n$ drinks ($n > 5$)?

Given two positive integers $n$ and $m$ and a function $f : \mathbb{Z} \times \mathbb{Z} \to \left\{0,1\right\}$ with the property that \begin{align*} f\left(i, j\right) = f\left(i+n, j\right) = f\left(i, j+m\right) \qquad \text{for all } \left(i, j\right) \in \mathbb{Z} \times \mathbb{Z} . \end{align*} Let $\left[k\right] = \left\{1,2,\ldots,k\right\}$ for each positive integer $k$. Let $a$ be the number of all $\left(i, j\right) \in \left[n\right] \times \left[m\right]$ satisfying \begin{align*} f\left(i, j\right) = f\left(i+1, j\right) = f\left(i, j+1\right) . \end{align*} Let $b$ be the number of all $\left(i, j\right) \in \left[n\right] \times \left[m\right]$ satisfying \begin{align*} f\left(i, j\right) = f\left(i-1, j\right) = f\left(i, j-1\right) . \end{align*} Prove that $a = b$.

There is a board with $2020$ squares in the bottom row and $2019$ in the top row, located as shown shown in the figure. [img]https://cdn.artofproblemsolving.com/attachments/f/3/516ad5485c399427638c3d1783593d79d83002.png[/img] In the bottom row the integers numbers from $ 1$ to $2020$ are placed in some order. Then in each box in the top row records the multiplication of the two numbers below it. How can they place the numbers in the bottom row so that the sum of the numbers in the top row be the smallest possible?

[u]Set 4[/u] [b]p16.[/b] Albert, Beatrice, Corey, and Dora are playing a card game with two decks of cards numbered $1-50$ each. Albert, Beatrice, and Corey draw cards from the same deck without replacement, but Dora draws from the other deck. What is the probability that the value of Corey’s card is the highest value or is tied for the highest value of all $4$ drawn cards? [b]p17.[/b] Suppose that $s$ is the sum of all positive values of $x$ that satisfy $2016\{x\} = x+[x]$. Find $\{s\}$. (Note: $[x]$ denotes the greatest integer less than or equal to $x$ and $\{x\}$ denotes $x - [x]$.) [b]p18.[/b] Let $ABC$ be a triangle such that $AB = 41$, $BC = 52$, and $CA = 15$. Let H be the intersection of the $B$ altitude and $C$ altitude. Furthermore let $P$ be a point on $AH$. Both $P$ and $H$ are reflected over $BC$ to form $P'$ and $H'$ . If the area of triangle $P'H'C$ is $60$, compute $PH$. [b]p19.[/b] A random integer $n$ is chosen between $1$ and $30$, inclusive. Then, a random positive divisor of $n, k$, is chosen. What is the probability that $k^2 > n$? [b]p20.[/b] What are the last two digits of the value $3^{361}$? [u]Set 5[/u] [b]p21.[/b] Let $f(n)$ denote the number of ways a $3 \times n$ board can be completely tiled with $1 \times 3$ and $1 \times 4$ tiles, without overlap or any tiles hanging over the edge. The tiles may be rotated. Find $\sum^9_{i=0} f(i) = f(0) + f(1) + ... + f(8) + f(9)$. By convention, $f(0) = 1$. [b]p22.[/b] Find the sum of all $5$-digit perfect squares whose digits are all distinct and come from the set $\{0, 2, 3, 5, 7, 8\}$. [b]p23.[/b] Mary is flipping a fair coin. On average, how many flips would it take for Mary to get $4$ heads and $2$ tails? [b]p24.[/b] A cylinder is formed by taking the unit circle on the $xy$-plane and extruding it to positive infinity. A plane with equation $z = 1 - x$ truncates the cylinder. As a result, there are three surfaces: a surface along the lateral side of the cylinder, an ellipse formed by the intersection of the plane and the cylinder, and the unit circle. What is the total surface area of the ellipse formed and the lateral surface? (The area of an ellipse with semi-major axis $a$ and semi-minor axis $b$ is $\pi ab$.) [b]p25.[/b] Let the Blair numbers be defined as follows: $B_0 = 5$, $B_1 = 1$, and $B_n = B_{n-1} + B_{n-2}$ for all $n \ge 2$. Evaluate $$\sum_{i=0}^{\infty} \frac{B_i}{51^i}= B_0 +\frac{B_1}{51} +\frac{B_2}{51^2} +\frac{B_3}{51^3} +...$$ [u]Estimation[/u] [b]p26.[/b] Choose an integer between $1$ and $10$, inclusive. Your score will be the number you choose divided by the number of teams that chose your number. [b]p27.[/b] $2016$ blind people each bring a hat to a party and leave their hat in a pile at the front door. As each partier leaves, they take a random hat from the ones remaining in a pile. Estimate the probability that at least $1$ person gets their own hat back. [b]p28.[/b] Estimate how many lattice points lie within the graph of $|x^3| + |y^3| < 2016$. [b]p29.[/b] Consider all ordered pairs of integers $(x, y)$ with $1 \le x, y \le 2016$. Estimate how many such ordered pairs are relatively prime. [b]p30.[/b] Estimate how many times the letter “e” appears among all Guts Round questions. PS. You should use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c3h2779594p24402189]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does there exist a set $S$ of $100$ points in a plane such that the center of mass of any $10$ points in $S$ is also a point in $S$?

Petya, Vasya and Tolya play a game on a $100\times 100$ board. They take turns (starting from Petya, then Vasya, then Tolya, then Petya, etc.) paint the boundary cells of the board (i.e., having a common side with the boundary of the board.) It is forbidden to paint the cell that is adjacent to the already painted one. In addition, it’s also forbidden to paint the cell which is symmetrical to the painted one, with respect to the center of the board. The player who can’t make the move loss. Can Vasya and Tolya, after agreeing, play so that Petya loses?

There is a $m \times (m-1)$ board. (i.e. there are $m+1$ horizontal lines and $m$ vertical lines) A stone is put on an intersection of the lowest horizontal line. Now two players move this stone with the following rules. (i) Each players move the stone to a neighboring intersection along a segment, by turns. (ii) A segment, which is already passed by the stone, cannot be used more. (iii) One who cannot move the stone anymore loses. Prove that there is a winning strategy for the former player.

The integer number $n > 1$ is given and a set $S \subset \{0, 1, 2, \ldots, n-1\}$ with $|S| > \frac{3}{4} n$. Prove that there exist integer numbers $a, b, c$ such that the remainders after the division by $n$ of the numbers: \[a, b, c, a+b, b+c, c+a, a+b+c\] belong to $S$.

$A$ writes, at his choice, $8$ ones and $8$ twos on a $4\times 4$ board. Then $B$ covers the board with $8$ dominoes and for each domino she finds the smaller of the two numbers that that domino covers. Finally, $A$ adds these $8$ numbers and the result is her score. What is the highest score $A$ can secure, no matter how $B$ plays? Clarification: A domino is a $1\times 2$ or $2\times 1$ rectangle that covers exactly two squares on the board.

There are $85$ soldiers with different heigth and age. Every day commander chooses random soldier and send him and also all soldiers that are taller and older than this soldier, or all soldiers that are lower and younger than this soldier to color grass. Prove that after $10$ days we can find two soldiers, that color grass at same days.

Let $m$ and $n$ be positive integers. A sequence of points $(A_0,A_1,\ldots,A_n)$ on the Cartesian plane is called [i]interesting[/i] if $A_i$ are all lattice points, the slopes of $OA_0,OA_1,\cdots,OA_n$ are strictly increasing ($O$ is the origin) and the area of triangle $OA_iA_{i+1}$ is equal to $\frac{1}{2}$ for $i=0,1,\ldots,n-1$. Let $(B_0,B_1,\cdots,B_n)$ be a sequence of points. We may insert a point $B$ between $B_i$ and $B_{i+1}$ if $\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}$, and the resulting sequence $(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)$ is called an [i]extension[/i] of the original sequence. Given two [i]interesting[/i] sequences $(C_0,C_1,\ldots,C_n)$ and $(D_0,D_1,\ldots,D_m)$, prove that if $C_0=D_0$ and $C_n=D_m$, then we may perform finitely many [i]extensions[/i] on each sequence until the resulting two sequences become identical.

We have an $8\times 8$ board. An [i]interior [/i] edge is an edge between two $1 \times 1$ cells. we cut the board into $1 \times 2$ dominoes. For an inner edge $k$, $N(k)$ denotes the number of ways to cut the board so that it cuts along edge $k$. Calculate the last digit of the sum we get if we add all $N(k)$, where $k$ is an inner edge.

prove that the number of permutations such that the order of each element is a multiple of $d$ is $\frac{n!}{(\frac{n}{d})!d^{\frac{n}{d}}} \prod_{i=0}^{\frac{n}{d}-1} (id+1)$. [i]proposed by Mohammad Mansouri[/i]

Let $ n \geq 3$ be an odd integer. We denote by $ [\minus{}n,n]$ the set of all integers greater or equal than $ \minus{}n$ and less or equal than $ n$. Player $ A$ chooses an arbitrary positive integer $ k$, then player $ B$ picks a subset of $ k$ (distinct) elements from $ [\minus{}n,n]$. Let this subset be $ S$. If all numbers in $ [\minus{}n,n]$ can be written as the sum of exactly $ n$ distinct elements of $ S$, then player $ A$ wins the game. If not, $ B$ wins. Find the least value of $ k$ such that player $ A$ can always win the game.

A natural number is written in each cell of an $8 \times 8$ board. It turned out that for any tiling of the board with dominoes, the sum of numbers in the cells of each domino is different. Can it happen that the largest number on the board is no greater than $32$? [i](N. Chernyatyev)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

Let $n$ be a positive integer, and let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word, consisting of just letters $a$ and/or $b$. Suppose that the minimal period $N$ of $W$ is greater than $2^n$. A finite nonempty word $U$ is said to [i]appear[/i] in $W$ if there exist indices $k \leq \ell$ such that $U=x_k x_{k+1} \ldots x_{\ell}$. A finite word $U$ is called [i]ubiquitous[/i] if the four words $Ua$, $Ub$, $aU$, and $bU$ all appear in $W$. Prove that there are at least $n$ ubiquitous finite nonempty words. [i]Proposed by Grigory Chelnokov, Russia[/i]

On a $1000\times 1000$-board we put dominoes, in such a way that each domino covers exactly two squares on the board. Moreover, two dominoes are not allowed to be adjacent, but are allowed to touch in a vertex. Determine the maximum number of dominoes that we can put on the board in this way. [i]Attention: you have to really prove that a greater number of dominoes is impossible. [/i]

Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$

The vertices of a regular $2002$-sided polygon are numbered $1$ through $2002$, clockwise. Given an integer $ n$, $1 \le n \le 2002$, color vertex $n$ blue, then, going clockwise, count$ n$ vertices starting at the next of $n$, and color $n$ blue. And so on, starting from the vertex that follows the last vertex that was colored, n vertices are counted, colored or uncolored, and the number $n$ is colored blue. When the vertex to be colored is already blue, the process stops. We denote $P(n)$ to the set of blue vertices obtained with this procedure when starting with vertex $n$. For example, $P(364)$ is made up of vertices $364$, $728$, $1092$, $1456$, $1820$, $182$, $546$, $910$, $1274$, $1638$, and $2002$. Determine all integers $n$, $1 \le n \le 2002$, such that $P(n)$ has exactly $14 $ vertices,

An integer $ n $ is called good if $ |n| $ is not a square of an integer. Find all integers $m$ with the following property: $m$ can be represented in infinite ways as a sum of three disctinct good numbers, the product of which is the square of an odd integer.