For a class of $20$ students several field trips were arranged. In each trip at least one student participated. Prove that there was a field trip such that each student who participated in it took part in at least $1/20$-th of all field trips.

The natural numbers $0, 1, 2, 3, . . .$ are written on the square table $2015\times 2015$ in a circular order (anti-clockwise) such that $0$ is in the center of the table. The rows and columns are labelled from bottom to top and from left to right respectively. (see figure below) 1. The number $2015$ is in which row and which column? 2. We are allowed to perform the following operations: First, we replace the number $0$ in the center by $14$, after that, each time, we can add $1$ to each of $12$ numbers on $12$ consecutive unit squares in a row, or $12$ consecutive unit squares in a column, or $12$ unit squares in a rectangle $3\times 4$. After a finite number of steps, can we make all numbers on the table are multiples of $2016$? [img]https://cdn.artofproblemsolving.com/attachments/c/d/223b32c0e3f58f62d0d40fa78c09a2cd035ed5.png[/img]

Four equal right-angled triangles are given. We are allowed to cut any triangle into two new ones along the altitude dropped on to the hypotenuse. This operation may be repeated with any of the triangles from the new set. Prove that after any number of such operations there will be congruent triangles among those obtained. (AV Shapovalov)

The vertices of a convex $n$-gon are colored so that adjacent vertices have different colors. Prove that if $n$ is odd, then the polygon can be divided into triangles with non-intersecting diagonals such that no diagonal has its endpoints the same color.

[b][u]BdMO National 2015 Secondary Problem 1.[/u][/b] A crime is committed during the hartal.There are four witnesses.The witnesses are logicians and make the following statement: Witness [b]One[/b] said exactly one of the four witnesses is a liar. Witness [b]Two[/b] said exactly two of the four witnesses is a liar. Witness [b]Three[/b] said exactly three of the four witnesses is a liar. Witness [b]Four[/b] said exactly four of the four witnesses is a liar. Assume that each of the statements is either true or false.How many of the winesses are liars?

Show that the edges of a strongly connected bipolar graph can be oriented in such a way that for any edge $ e$ there is a simple directed path from pole $ p$ to pole $ q$ containing $ e$. (A strongly connected bipolar graph is a finite connected graph with two special vertices $ p$ and $ q$ having the property that there are no points $ x,y,x \not \equal{} y$, such that all paths from $ x$ to $ p$ as well as all paths from $ x$ to $ q$ contain $ y$.) [i]A. Adam[/i]

For each positive integer $k,$ let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n,$ such that all the differences \[t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)\] are divisible by 4. [i]Proposed by Gerhard Wöginger, Austria[/i]

Consider an 8x8 board divided in 64 unit squares. We call [i]diagonal[/i] in this board a set of 8 squares with the property that on each of the rows and the columns of the board there is exactly one square of the [i]diagonal[/i]. Some of the squares of this board are coloured such that in every [i]diagonal[/i] there are exactly two coloured squares. Prove that there exist two rows or two columns whose squares are all coloured.

Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible? $\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$

Let $A = \{1,2,..., 100\}$ and $B$ is a subset of $A$ having $48$ elements. Show that $B$ has two distint elements $x$ and $y$ whose sum is divisible by $11$.

Ten vertices of a regular $20$-gon $A_1A_2....A_{20}$ are painted black and the other ten vertices are painted blue. Consider the set consisting of diagonal $A_1A_4$ and all other diagonals of the same length. 1. Prove that in this set, the number of diagonals with two black endpoints is equal to the number of diagonals with two blue endpoints. 2. Find all possible numbers of the diagonals with two black endpoints.

The [i]triangular domino[/i] is a game that uses the tokens shown below, with equilateral triangle shape with side $ 1$. The idea of the game is to construct an equilateral triangle with side $n$, no gaps, following the rules of the domino or classic. $\bullet$ Show that the sum $S$ of the values corresponding to the edges that are part of the sides of the greater triangle, it depends only on n, and not on the way in which the tokens are paired. $\bullet$ For each value of $n$, calculate $S$. [img]https://cdn.artofproblemsolving.com/attachments/e/9/898664fac380725a7398dfe470298a90b8c69b.png[/img]

Every cell of a $2017\times 2017$ grid is colored either black or white, such that every cell has at least one side in common with another cell of the same color. Let $V_1$ be the set of all black cells, $V_2$ be the set of all white cells. For set $V_i (i=1,2)$, if two cells share a common side, draw an edge with the centers of the two cells as endpoints, obtaining graphs $G_i$. If both $G_1$ and $G_2$ are connected paths (no cycles, no splits), prove that the center of the grid is one of the endpoints of $G_1$ or $G_2$.

Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector). At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy? [i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]

Assume that the connected graph $G$ has $n$ vertices all with degree at least three. Prove that there exists a spanning tree of $G$ with more than $\frac{2}{9}n$ leaves.

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: [list=1] [*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell. [/list] At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Let $Q_n$ be the set of all $n$-tuples $x=(x_1,\ldots,x_n)$ with $x_i \in \{0,1,2 \}$, $i=1,2,\ldots,n$. A triple $(x,y,z)$ (where $x=(x_1,x_2,\ldots,x_n)$, $y=(y_1,y_2,\ldots,y_n)$, $z=(z_1,z_2,\ldots,z_n)$) of distinct elements of $Q_n$ is called a [i]good[/i] triple, if there exists at least one $i \in \{1,2, \ldots, n \}$, for which $\{x_i,y_i,z_i \}=\{0,1,2 \}$. A subset $A$ of $Q_n$ will be called a [i]good[/i] subset, if any three elements of $A$ form a [i]good[/i] triple. Prove that every [i]good[/i] subset of $Q_n$ contains at most $2 \cdot \left(\frac{3}{2}\right)^n$ elements.

12. What is the sum of all possible $\left(\begin{array}{l}i \\ j\end{array}\right)$ subject to the restrictions that $i \geq 10, j \geq 0$, and $i+j \leq 20$ ? Count different $i, j$ that yield the same value separately - for example, count both $\left(\begin{array}{c}10 \\ 1\end{array}\right)$ and $\left(\begin{array}{c}10 \\ 9\end{array}\right)$.

A tennis tournament has at least three participants. Every participant plays exactly one match against every other participant. Moreover, every participant wins at least one of the matches he plays. (Draws do not occur in tennis matches.) Show that there are three participants $A, B $ and $C$ for which the following holds: $A$ wins against $B, B$ wins against $C$, and $C$ wins against $A$.

There are $ n ( \ge 4 ) $ people and some people shaked hands each other. Two people can shake hands at most 1 time. For arbitrary four people $ A, B, C, D$ such that $ (A,B), (B,C), (C,D) $ shaked hands, then one of $ (A,C), (A,D), (B,D) $ shaked hand each other. Prove the following statements. (a) Prove that $ n $ people can be divided into two groups, $ X, Y ( \ne \emptyset )$ , such that for all $ (x,y) $ where $ x \in X $ and $ y \in Y $, $ x $ and $ y $ shaked hands or $ x $ and $ y $ didn't shake hands. (b) There exist two people $ A , B $ such that the set of people who are not $ A $ and $ B $ that shaked hands with $ A $ is same wiith the set of people who are not $ A $ and $ B $ that shaked hands with $ B $.

A [i]beautiful configuration[/i] of points is a set of $n$ colored points, such that if a triangle with vertices in the set has an angle of at least $120$ degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of $n$.

How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one? [i]Proposed by Germany, FR.[/i]

At a dance there are $12$ men, numbered $1$ to $12$, and $12$ women numbered $1$ to $12$. Each man is assigned a “secret friend” among the $11$ others. They all danced all the pieces. In the first piece each man danced with the woman who has the same number. From then on, each man danced the new piece with the woman who had danced the piece earlier with his secret friend. In the third piece the couples were: [img]https://cdn.artofproblemsolving.com/attachments/c/d/f5ea0931e5751739c1ba556f84ab5736f2d11a.png[/img] Find the number of each man's secret friend.

An arrangement of chips in the squares of $ n\times n$ table is called [i]sparse[/i] if every $ 2\times 2$ square contains at most 3 chips. Serge put chips in some squares of the table (one in a square) and obtained a sparse arrangement. He noted however that if any chip is moved to any free square then the arrangement is no more sparce. For what $ n$ is this possible? [i]Proposed by S. Berlov[/i]

$40$ cells were marked on an infinite chessboard. Is it always possible to find a rectangle that contains $20$ marked cells? M. Evdokimov