For which positive integers $n \ge 3$ is it possible to mark $n$ points of a plane in such a way that, starting from one marked point and moving on each step to the marked point which is the second closest to the current point, one can walk through all the marked points and return to the initial one? For each point, the second closest marked point must be uniquely determined.

There are $b$ boys and $g$ girls, with $g \ge 2b-1$, at presence at a party. Each boy invites a girl for the first dance. Prove that this can be done in such a way that either a boy is dancing with a girl he knows or all the girls he knows are not dancing.

Let $m$ be a positive integer with $m \leq 2024$. Ana and Banana play a game alternately on a $1\times2024$ board, with squares initially painted white. Ana starts the game. Each move by Ana consists of choosing any $k \leq m$ white squares on the board and painting them all green. Each Banana play consists of choosing any sequence of consecutive green squares and painting them all white. What is the smallest value of $m$ for which Ana can guarantee that, after one of her moves, the entire board will be painted green?

Divide the numbers $$1, 2,3, 4,5$$ into two arbitrarily chosen sets. Prove that one of the sets contains two numbers and their difference.

We may permute the rows and the columns of the table below. How may different tables can we generate? 1 2 3 4 5 6 7 7 1 2 3 4 5 6 6 7 1 2 3 4 5 5 6 7 1 2 3 4 4 5 6 7 1 2 3 3 4 5 6 7 1 2 2 3 4 5 6 7 1

Let $n<k$ be two natural numbers and suppose that Sepehr has $n$ chemical elements , $2k$ grams from each , divided arbitrarily in $2k$ cups.Find the smallest number $b$ such that there is always possible for Sepehr to choose $b$ cups , containing at least $2$ grams from each element in total. [i]Proposed by Josef Tkadlec & Morteza Saghafian[/i]

A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies \[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \] for all $k=1,2,\ldots 9$. Find the number of interesting numbers.

$(CZS 4)$ Let $K_1,\cdots , K_n$ be nonnegative integers. Prove that $K_1!K_2!\cdots K_n! \ge \left[\frac{K}{n}\right]!^n$, where $K = K_1 + \cdots + K_n$

What is the minimum $n$ so that grid $nxn$ can be covered with equal number of 2x2 squares and angle triminoes (2x2 without one square)

In each cell of a $2019\times 2019$ board is written the number $1$ or the number $-1$. Prove that for some positive integer $k$ it is possible to select $k$ rows and $k$ columns so that the absolute value of the sum of the $k^2$ numbers in the cells at the intersection of the selected rows and columns is more than $1000$. [i]Folklore[/i]

Each cell of a $29 \times 29$ table contains one of the integers $1, 2, 3, \ldots , 29$, and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.

Each of the $n$ students in a class sent a card to each of his $m$ colleagues. Prove that if $2m + 1 > n$, then at least two students sent cards to each other.

There is a regular hexagon that is cut direct to $6n^2$ equilateral triangles (Fig.). There are arranged $2n$ rooks, neither of which beats each other (the rooks hit in directions parallel to sides of the hexagon). Prove that if we consider chess coloring all $6n^2$ equilateral triangles, then the number of rooks that stand on black triangles will be equal to the number of rooks standing on white triangles. [img]https://cdn.artofproblemsolving.com/attachments/d/0/43ce6c5c966f60a8ec893d5d8cd31e33c43fc0.png[/img] [hide=original wording] Є правильний шестикутник, що розрізаний прямими на 6n^2 правильних трикутників (рис. 2). У них розставлені 2n тур, ніякі дві з яких не б'ють одна одну (тура б'є в напрямках, що паралельні до сторін шестикутника). Доведіть, що якщо розглянути шахове розфарбування всіх 6n^2 правильних трикутників, то тоді кількість тур, що стоять на чорних трикутниках, буде рівна кількості тур, що стоять на білих трикутниках. [/hide]

Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches. What is the largest possible number of matches?

We have $1000$ balls in $40$ different colours, $25$ balls of each colour. Determine the smallest $n$ for which the following holds: if you place the $1000$ balls in a circle, in any arbitrary way, then there are always $n$ adjacent balls which have at least $20$ different colours.

How many ways are there to divide $10$ candies between $3$ Berkeley students and $4$ Stanford students, if each Berkeley student must get at least one candy? All students are distinguishable from each other; all candies are indistinguishable.

[i](a)[/i] Determine if the set $ \{1,2,\ldots,96\}$ can be partitioned into 32 sets of equal size and equal sum. [i](b)[/i] Determine if the set $ \{1,2,\ldots,99\}$ can be partitioned into 33 sets of equal size and equal sum.

There are $256$ players in a tennis tournament who are ranked from $1$ to $256$, with $1$ corresponding to the highest rank and $256$ corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability $\frac{3}{5}$. In each round of the tournament, the player with the highest rank plays against the player with the second highest rank, the player with the third highest rank plays against the player with the fourth highest rank, and so on. At the end of the round, the players who win proceed to the next round and the players who lose exit the tournament. After eight rounds, there is one player remaining and they are declared the winner. Determine the expected value of the rank of the winner.

Prove that among $20$ consecutive positive integers there is an integer $d$ such that for every positive integer $n$ the following inequality holds $$n \sqrt{d} \left\{n \sqrt {d} \right \} > \dfrac{5}{2}$$ where by $\left \{x \right \}$ denotes the fractional part of the real number $x$. The fractional part of the real number $x$ is defined as the difference between the largest integer that is less than or equal to $x$ to the actual number $x$. [i](Serbia)[/i]

Pawns and rooks are placed on a $2019\times 2019$ chessboard, with at most one piece on each of the $2019^2$ squares. A rook [i]can see[/i] another rook if they are in the same row or column and all squares between them are empty. What is the maximal number $p$ for which $p$ pawns and $p+2019$ rooks can be placed on the chessboard in such a way that no two rooks can see each other?

Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied: a) For any $x \in M, n \in \mathbb{Z}$, one has that $x+n \in \mathbb{M}$. b) For any $x \in M$, one has that $-x \in M$. c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$. For any real $x$, let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$. Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$, then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Suppose $a_1,a_2,\cdots,a_8$ are eight distinct integers from $\{1,2,\cdots,16,17\}$. Show that there is an integer $k > 0$ such that the equation $a_i - a_j = k$ has at least three different solutions. Also, find a specific set of 7 distinct integers from $\{1,2,\ldots,16,17\}$ such that the equation $a_i - a_j = k$ does not have three distinct solutions for any $k > 0$.

We divide a convex $2009$-gon in triangles using non-intersecting diagonals. One of these diagonals is colored green. It is allowed the following operation: for two triangles $ABC$ and $BCD$ from the dividing/separating with a common side $BC$ if the replaced diagonal was green it loses its color and the replacing diagonal becomes green colored. Prove that if we choose any diagonal in advance it can be colored in green after applying the operation described finite number of times.

In a factory, square tables of $ 40 \times 40$ are tiled with four tiles of size $ 20 \times 20$. All tiles are the same and decorated in the same way with an asymmetric pattern such as the letter $ J$. How many different types of tables can be produced in this way?