For $2n$ numbers in a row, Bob could perform the following operation: $$S_i=(a_1,a_2,\ldots,a_{2n})\mapsto S_{i+1}=(a_1,a_3,\ldots,a_{2n-1},a_2,a_4,\ldots,a_{2n}).$$ Let $T$ be the order of this operation. In other words, $T$ is the smallest positive integer such that $S_i=S_{i+T}$. Prove that $T<2n$.

The sequence $1, 2, \dots, 2023, 2024$ is written on a whiteboard. Every second, Megavan chooses two integers $a$ and $b$, and four consecutive numbers on the whiteboard. Then counting from the left, he adds $a$ to the 1st and 3rd of those numbers, and adds $b$ to the 2nd and 4th of those numbers. Can he achieve the sequence $2024, 2023, \dots, 2, 1$ in a finite number of moves? [i](Proposed by Avan Lim Zenn Ee)[/i]

An urn contains balls of $k$ different colors; there are $n_i$ balls of $i-th$ color. Balls are selected at random from the urn, one by one, without replacement, until among the selected balls $m$ balls of the same color appear. Find the greatest number of selections.

Let be given $ n \ge 2$ lines on a plane, not all concurrent and no two parallel. Prove that there is a point which belongs to exactly two of the given lines.

Consider the set of natural numbers $ U = \{1,2,3, ..., 6024 \} $ Prove that for any partition of the $ U $ in three subsets with $ 2008 $ elements each, we can choose a number in each subset so that one of the numbers is the sum of the other two numbers.

Let $\alpha \geq 1$ be a real number. Hephaestus and Poseidon play a turn-based game on an infinite grid of unit squares. Before the game starts, Poseidon chooses a finite number of cells to be [i]flooded[/i]. Hephaestus is building a [i]levee[/i], which is a subset of unit edges of the grid (called [i]walls[/i]) forming a connected, non-self-intersecting path or loop*. The game then begins with Hephaestus moving first. On each of Hephaestus’s turns, he adds one or more walls to the levee, as long as the total length of the levee is at most $\alpha n$ after his $n$th turn. On each of Poseidon’s turns, every cell which is adjacent to an already flooded cell and with no wall between them becomes flooded as well. Hephaestus wins if the levee forms a closed loop such that all flooded cells are contained in the interior of the loop — hence stopping the flood and saving the world. For which $\alpha$ can Hephaestus guarantee victory in a finite number of turns no matter how Poseidon chooses the initial cells to flood? ----- [size=75]*More formally, there must exist lattice points $\mbox{\footnotesize \(A_0, A_1, \dotsc, A_k\)}$, pairwise distinct except possibly $\mbox{\footnotesize \(A_0 = A_k\)}$, such that the set of walls is exactly $\mbox{\footnotesize \(\{A_0A_1, A_1A_2, \dotsc , A_{k-1}A_k\}\)}$. Once a wall is built it cannot be destroyed; in particular, if the levee is a closed loop (i.e. $\mbox{\footnotesize \(A_0 = A_k\)}$) then Hephaestus cannot add more walls. Since each wall has length $\mbox{\footnotesize \(1\)}$, the length of the levee is $\mbox{\footnotesize \(k\)}$.[/size] [i]Nikolai Beluhov[/i]

A circle is somehow divided by $3k$ points into $k$ arcs of lengths $1, 2$ and $3$ each. Prove that two of these points are always diametrically opposite.

Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|, |b - d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1, 1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_0, y_0)= (1, 1)$, $(x_1, y_1), (x_2, y_2)$, $\ldots, (x_n, y_n)$ such that, for all $i = 1, 2, \ldots, n$, $1 \leq x_i , y_i \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_i, y_i)$. In this case, each cell $(x_i, y_i)$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.

(Game) At the beginning of the game the organisers place $4$ piles of paper disks onto the table. The player who is in turn takes away a pile, then divides one of the remaining piles into two nonempty piles. Whoever is unable to move, loses. [i]Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.[/i]

Consider the regular $101$-gon. A line $l$ does not contain any vertex of this polygon. Prove that line $l$ intersects even number of the diagonals of this polygon.

In the rectangular coordinate system every point with integer coordinates is called laticeal point. Let $P_n(n, n + 5)$ be a laticeal point and denote by $f(n)$ the number of laticeal points on the open segment $(OP_n)$, where the point $0(0,0)$ is the coordinates system origine. Calculate the number $f(1) +f(2) + f(3) + ...+ f(2002) + f(2003)$.

A difficult mathematical competition consisted of a Part I and a Part II with a combined total of $28$ problems. Each contestant solved $7$ problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.

$n$ points are given on a circle $\omega$. There is a circle with radius smaller than $\omega$ such that all these points lie inside or on the boundary of this circle. Prove that we can draw a diameter of $\omega$ with endpoints not belonging to the given points such that all the $n$ given points remain in one side of the diameter.

For a nonzero integer $k$, denote by $\nu_2(k)$ the maximal nonnegative integer $t$ such that $2^t \mid k$. Given are $n (\ge 2)$ pairwise distinct integers $a_1, a_2, \ldots, a_n$. Show that there exists an integer $x$, distinct from $a_1, \ldots, a_n$, such that among $\nu_2(x - a_1), \ldots, \nu_2(x - a_n)$ there are at least $n/4$ odd numbers and at least $n/4$ even numbers.

Consider a table with three rows and eleven columns. There are zeroes prefilled in the cell of the first row and the first column and in the cell of the second row and the last column. Determine the least real number $\alpha$ such that the table can be filled with non-negative numbers and the following conditions hold simultaneously: (1) the sum of numbers in every column is one, (2) the sum of every two neighboring numbers in the first row is at most one, (3) the sum of every two neighboring numbers in the second row is at most one, (4) the sum of every two neighboring numbers in the third row is at most $\alpha$.

Two sets of intervals $A ,B$ on the line are given. The set $A$ contains $2m-1$ intervals, every two of which have an interior point in common. Moreover, every interval from $A$ contains at least two disjoint intervals from $B$. Show that there exists an interval in $B$ which belongs to at least $m$ intervals from $A$ .

Let $S= \{ a_1, \ldots, a_n\}$ be a set of $n\geq 1$ positive real numbers. For each nonempty subset of $S$ the sum of its elements is written down. Show that all written numbers can be divided into $n$ classes such that in each class the ratio of the greatest number to the smallest number is not greater than $2$.

Let $n$ be a positive odd integer. $C$ is a set consists of integral points on a plane, which is defined by \[ C = \{(i, j): i, j = 0, 1, \dots, 2n-1\} \] and forms a $2n \times 2n$ array. On every point there is a Guinea pig, which is facing toward one of the following directions: [i]positive/negative $x$-axis[/i], or [i]positive/negative $y$-axis[/i]. Jeff wants to keep $n^2+1$ of the Guinea pigs on the plane and remove all the others. After that, the Guinea pigs on the plane will move as the following: 1. In every round, the Guinea pigs move toward by an unit, and keep facing the same direction. 2. If a Guinea pig move to a point $(i, j)$ which is [i]not[/i] in $C$, it will further move to another point $(p, q)$ in $C$, such that $p \equiv i \pmod {2n}$ and $q \equiv j \pmod {2n}$. [i](For example, if a Guinea pig move from $(2, 0)$ to $(2, -1)$, it will then further move to $(2, 2n-1)$.)[/i] The next round begins after all the Guinea pigs settle up. Jeff's goal is to keep the appropriate Guinea pigs on the plane, so that in every single round, any two Guinea pigs will never move to the same endpoint, and will never move to the startpoints[i](in that round)[/i] of each other simultaneously. Prove that Jeff can always succeed wherever the Guinea pigs initially face. [i]Proposed by Weijiun Kao[/i] Edit: By the way, it can be proven that the number $n^2+1$ is optimal, i.e. if the Guinea pigs face appropriately, Jeff can only keep at most $n^2+1$ of them on the plane to avoid any collision.

[b]N[/b]ew this year at HMNT: the exciting game of RNG baseball! In RNG baseball, a team of infinitely many people play on a square field, with a base at each vertex; in particular, one of the bases is called the home base. Every turn, a new player stands at home base and chooses a number n uniformly at random from $\{0, 1, 2, 3, 4\}$. Then, the following occurs: • If $n>0$, then the player and everyone else currently on the field moves (counterclockwise) around the square by n bases. However, if in doing so a player returns to or moves past the home base, he/she leaves the field immediately and the team scores one point. • If $n=0$ (a strikeout), then the game ends immediately; the team does not score any more points. What is the expected number of points that a given team will score in this game?

[list] [*] A test was conducted in class. It is known that at least $\frac{2}{3}$ of the problems were hard. Each such problems were not solved by at least $\frac{2}{3}$ of the students. It is also known that at least $\frac{2}{3}$ of the students passed the test. Each such student solved at least $\frac{2}{3}$ of the suggested problems. Is this possible? [*] Previous problem with $\frac{2}{3}$ replaced by $\frac{3}{4}$. [*] Previous problem with $\frac{2}{3}$ replaced by $\frac{7}{10}$.[/list]

In each cell of an $n\times n$ board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off). Show that if after a certain number of moves, at least one light is on, then at this moment at least $n$ lights are on.

There are $n$ boyfriend-girlfriend pairs at a party. Initially all the girls sit at a round table. For the first dance, each boy invites one of the girls to dance with.After each dance, a boy takes the girl he danced with to her seat, and for the next dance he invites the girl next to her in the counterclockwise direction. For which values of $n$ can the girls be selected in such a way that in every dance at least one boy danced with his girlfriend, assuming that there are no less than $n$ dances?

In a simple graph $G$, we call $t$ pairwise adjacent vertices a $t$[i]-clique[/i]. If a vertex is connected with all other vertices in the graph, we call it a [i]central[/i] vertex. Given are two integers $n,k$ such that $\dfrac {3}{2} \leq \dfrac{1}{2} n < k < n$. Let $G$ be a graph on $n$ vertices such that [b](1)[/b] $G$ does not contain a $(k+1)$-[i]clique[/i]; [b](2)[/b] if we add an arbitrary edge to $G$, that creates a $(k+1)$-[i]clique[/i]. Find the least possible number of [i]central[/i] vertices in $G$.

The water city of Platense consists of many platforms and bridges between them. Each bridge connects two platforms and there is not two bridges connecting the same two platforms. The mayor wants to switch some bridges by a series of moves in the following way: if there are three platforms $A,B,C$ and bridges $AB$ and $AC$ ([b]no[/b] bridge $BC$), he can switch bridge $AB$ to a bridge $BC$. A configuration of bridges is [i]good[/i] if it is possible to go to any platfom from any platform using only bridges. Starting in a good configuration, prove that the mayor can reach any other good configuration, whose the quantity of bridges is the same, using the allowed moves.

In the expansion of $(2x +3y)^{20}$, find the number of coefficients divisible by $144$. [i]Proposed by Hannah Shen[/i]