$16$ chess players held a tournament among themselves: every two chess players played exactly one game. For victory in the party was given $1$ point, for a draw $0.5$ points, for defeat $0$ points. It turned out that exactly 15 chess players shared the first place. How many points could the sixteenth chess player score?

There is a $2021 \times 2023$ board that has a white piece in the central square, on which Mich and Moka are going to play in turns. First Mich places a green token on any free space so that it is not in the same row or column as the white token, then Moka places a red token on any free space so that it is not in the same row or column as the white token. white or green. From now on, Mich will place green tokens and Moka will place red tokens alternately according to the following rules: $\bullet$ For the placed piece there must be another piece of the same color in its row or column, such that there is no other piece between both pieces. $\bullet$ If there is at least one box that meets the previous rule, then it is mandatory to place a token. When a token is placed, it changes all the tokens that are on squares adjacent to it to the same color. The game ends when one of the players can no longer place tiles. If when the game ends the board has more green tiles then Mich wins, and if it has more red tiles then Moka wins. Determine if either player has a winning strategy.

Albert and Barbara play the following game. On a table there are $1999$ sticks, and each player in turn removes some of them: at least one stick, but at most half of the currently remaining sticks. The player who leaves just one stick on the table loses the game. Barbara moves first. Decide which player has a winning strategy and describe that strategy.

A journalist wants to report on the island of scoundrels and knights, where all inhabitants are either scoundrels (and they always lie) or knights (and they always tell the truth). The journalist interviews each inhabitant exactly once and gets the following answers: $A_1$: On this island there is at least one scoundrel, $A_2$: On this island there are at least two scoundrels, $...$ $A_{n-1}$: On this island there are at least $n-1$ scoundrels, $A_n$: On this island everybody is a scoundrel. Can the journalist decide whether there are more scoundrels or more knights?

Mom brought Andriy and Olesya $4$ balls with the numbers $1, 2, 3$ and $4$ written on them (one on each ball). She held $2$ balls in each hand and did not know which numbers were written on the balls in each hand. The mother asked Andriy to take a ball with a higher number from each hand, and then to keep the ball with the lower number from the two balls he took. After that, she asked Olesya to take two other balls, and out of these two, keep the ball with the higher number. Does the mother know with certainty, which child has the ball with the higher number? [i]Proposed by Bogdan Rublov[/i]

The numbers $1, 2, 3, ... , n$ are written on a blackboard (where $n \ge 3$). A move is to replace two numbers by their sum and non-negative difference. A series of moves makes all the numbers equal $k$. Find all possible $k$

A and B play a game on a square board consisting of $n \times n$ white tiles, where $n \ge 2$. A moves first, and the players alternate taking turns. A move consists of picking a square consisting of $2\times 2$ or $3\times 3$ white tiles and colouring all these tiles black. The first player who cannot find any such squares has lost. Show that A can always win the game if A plays the game right.

Let $n\ge 3$ be an integer. Lucas and Matías play a game in a regular $n$-sided polygon with a vertex marked as a trap. Initially Matías places a token at one vertex of the polygon. In each step, Lucas says a positive integer and Matías moves the token that number of vertices clockwise or counterclockwise, at his choice. a) Determine all the $n\ge 3$ such that Matías can locate the token and move it in such a way as to never fall into the trap, regardless of the numbers Lucas says. Give the strategy to Matías. b) Determine all the $n\ge 3$ such that Lucas can force Matías to fall into the trap. Give the strategy to Lucas. Note. The two players know the value of $n$ and see the polygon.

Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees? [i]Proposed by Seyed Reza Hosseini[/i]

A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of $x^3$ is already fixed at $1$. Can the first player make his choices so that the final cubic has three distinct integer roots?

Anna and Brian play a game where they put the domino tiles (of size $2 \times 1$) in a boards composed of $n \times 1$ boxes. Tiles must be placed so that they cover exactly two boxes. Players take turnslaying each tile and the one laying last tile wins. They play once for each $n$, where $n = 2, 3,\dots,2007$. Show that Anna wins at least $1505$ of the games if she always starts first and they both always play optimally, ie if they do their best to win in every move.

There is a board with $n$ rows and $4$ columns, and white, yellow and light blue chips. Player $A$ places four tokens on the first row of the board and covers them so Player $B$ doesn't know them. How should player $B$ do to fill the minimum number of rows with chips that will ensure that in any of the rows he will have at least three hits? Clarification: A hit by player $B$ occurs when he places a token of the same color and in the same column as $A$.

The cells of a $8 \times 8$ table are initially white. Alice and Bob play a game. First Alice paints $n$ of the fields in red. Then Bob chooses $4$ rows and $4$ columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of $n$ such that Alice can win the game no matter how Bob plays.

Given an initial integer $ n_0 > 1$, two players, $ {\mathcal A}$ and $ {\mathcal B}$, choose integers $ n_1$, $ n_2$, $ n_3$, $ \ldots$ alternately according to the following rules : [b]I.)[/b] Knowing $ n_{2k}$, $ {\mathcal A}$ chooses any integer $ n_{2k \plus{} 1}$ such that \[ n_{2k} \leq n_{2k \plus{} 1} \leq n_{2k}^2. \] [b]II.)[/b] Knowing $ n_{2k \plus{} 1}$, $ {\mathcal B}$ chooses any integer $ n_{2k \plus{} 2}$ such that \[ \frac {n_{2k \plus{} 1}}{n_{2k \plus{} 2}} \] is a prime raised to a positive integer power. Player $ {\mathcal A}$ wins the game by choosing the number 1990; player $ {\mathcal B}$ wins by choosing the number 1. For which $ n_0$ does : [b]a.)[/b] $ {\mathcal A}$ have a winning strategy? [b]b.)[/b] $ {\mathcal B}$ have a winning strategy? [b]c.)[/b] Neither player have a winning strategy?

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )