$n$ cars are racing. At first they have a particular order. At each moment a car may overtake another car. No two overtaking actions occur at the same time, and except moments a car is passing another, the cars always have an order. A set of overtaking actions is called "small" if any car overtakes at most once. A set of overtaking actions is called "complete" if any car overtakes exactly once. If $F$ is the set of all possible orders of the cars after a small set of overtaking actions and $G$ is the set of all possible orders of the cars after a complete set of overtaking actions, prove that \[\mid F\mid=2\mid G\mid\] (20 points) [i]Proposed by Morteza Saghafian[/i]

This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant $X$, let $t(X)$ be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true: $1)$ For any two friends $X'$ and $X''$, we have $t(X') \neq t(X''),$ $2)$ For every contestant $X$, the set $\{ t(Y) : Y \text{ is a friend of } X \}$ consists of consecutive integers. The organizers want to distribute the contestants into contest halls in such a way that no two friends are in the same hall. What is the minimal number of halls they need?

Let $n,s,$ and $t$ be positive integers and $0<\lambda<1.$ A simple graph on $n$ vertices with at least $\lambda n^2$ edges is given. We say that $(x_1,\ldots,x_s,y_1,\ldots,y_t)$ is a [i]good intersection[/i] if letters $x_i$ and $y_j$ denote not necessarily distinct vertices and every $x_iy_j$ is an edge of the graph $(1\leq i\leq s,$ $1\leq j\leq t).$ Prove that the number of good insertions is at least $\lambda^{st}n^{s+t}.$ [i]Proposed by Kada Williams, Cambridge[/i]

A set consisting of $n$ points of a plane is called an isosceles $n$-point if any three of its points are located in vertices of an isosceles triangle. Find all natural numbers for which there exist isosceles $n$-points.

Investigate whether a chess knight can traverse a $4 \times 4$ mini-chessboard so that it reaches each of the $16$ squares only once. Note: the drawing below shows the endpoints of the eight possible moves of the knight $(C)$ on a chessboard of size $8 \times 8$. [asy] unitsize(0.4 cm); int i; fill(shift((2,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((4,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((1,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((5,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((1,5))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((5,5))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((2,6))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((4,6))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); for (i = 0; i <= 8; ++i) { draw((i,0)--(i,8)); draw((0,i)--(8,i)); } label("C", (3.5,4.5), fontsize(8)); [/asy]

A board $n$×$n$ is coloured black and white like a chessboard. The following steps are permitted: Choose a rectangle inside the board (consisting of entire cells)whose side lengths are both odd or both even, but not both equal to $1$, and invert the colours of all cells inside the rectangle. Determine the values of $n$ for which it is possible to make all the cells have the same colour in a finite number of such steps.

Ali and Shayan are playing a turn-based game on an infinite grid. Initially, all cells are white. Ali starts the game, and in the first turn, he colors one unit square black. In the following turns, each player must color a white square that shares at least one side with a black square. The game continues for exactly 2808 turns, after which each player has made 1404 moves. Let $A$ be the set of black cells at the end of the game. Ali and Shayan respectively aim to minimize and maximise the perimeter of the shape $A$ by playing optimally. (The perimeter of shape $A$ is defined as the total length of the boundary segments between a black and a white cell.) What are the possible values of the perimeter of $A$, assuming both players play optimally?

Is it possible to place $1965$ points in a square with side $1$ so that any rectangle of area $1/200$ with sides parallel to the sides of the square contains at least one of these points inside?

Consider a regular 2n-gon $ P$,$A_1,A_2,\cdots ,A_{2n}$ in the plane ,where $n$ is a positive integer . We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ , if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$ .We color the sides of $P$ in 3 different colors (ignore the vertices of $P$,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover ,from every point in the plane external to $P$ , points of most 2 different colors on $P$ can be seen .Find the number of distinct such colorings of $P$ (two colorings are considered distinct if at least one of sides is colored differently). [i]Proposed by Viktor Simjanoski, Macedonia[/i] JBMO 2017, Q4

$ n\geq 4 $ points are given in a plane such that any 3 of them are not collinear. Prove that a triangle exist such that all the points are in its interior and there is exactly one point laying on each side.

Given positive integers $n$ and $k$, $n > k^2 >4.$ In a $n \times n$ grid, a $k$[i]-group[/i] is a set of $k$ unit squares lying in different rows and different columns. Determine the maximal possible $N$, such that one can choose $N$ unit squares in the grid and color them, with the following condition holds: in any $k$[i]-group[/i] from the colored $N$ unit squares, there are two squares with the same color, and there are also two squares with different colors.

Annie has a permutation $(a_1, a_2, \dots ,a_{2019})$ of $S=\{1,2,\dots,2019\}$, and Yannick wants to guess her permutation. With each guess Yannick gives Annie an $n$-tuple $(y_1, y_2, \dots, y_{2019})$ of integers in $S$, and then Annie gives the number of indices $i\in S$ such that $a_i=y_i$. (a) Show that Yannick can always guess Annie's permutation with at most $1200000$ guesses. (b) Show that Yannick can always guess Annie's permutation with at most $24000$ guesses. [i]Yannick Yao[/i]

We have 19 triminos (2 x 2 squares without one unit square) and infinite amount of 2 x 2 squares. Find the greatest odd number $n$ for which a square $n$ x $n$ can be covered with the given figures.

A square board with a size of $2020 \times 2020$ is divided into $2020^2$ small squares of size $1 \times 1$. Each of these small squares will be coloured black or white. Determine the number of ways to colour the board such that for every $2\times 2$ square, which consists of $4$ small squares, contains $2$ black small squares and $2$ white small squares.

Given positive integer $n (n \geq 2)$, find the largest positive integer $\lambda$ satisfying : For $n$ bags, if every bag contains some balls whose weights are all integer powers of $2$ (the weights of balls in a bag may not be distinct), and the total weights of balls in every bag are equal, then there exists a weight among these balls such that the total number of balls with this weight is at least $\lambda$.

Let $x_1,\ldots,x_n$ be a sequence with each term equal to $0$ or $1$. Form a triangle as follows: its first row is $x_1,\ldots,x_n$ and if a row if $a_1, a_2, \ldots, a_m$, then the next row is $a_1 + a_2, a_2 + a_3, \ldots, a_{m-1} + a_m$ where the addition is performed modulo $2$ (so $1+1=0$). For example, starting from $1$, $0$, $1$, $0$, the second row is $1$, $1$, $1$, the third one is $0$, $0$ and the fourth one is $0$. A sequence is called good it is the same as the sequence formed by taking the last element of each row, starting from the last row (so in the above example, the sequence is $1010$ and the corresponding sequence from last terms is $0010$ and they are not equal in this case). How many possibilities are there for the sequence formed by taking the first element of each row, starting from the last row, which arise from a good sequence?

Initially $A$ selects a graph with \( 2221 \) vertices such that each vertex is incident to at least one edge. Then $B$ deletes some of the edges (possibly none) from the chosen graph. Finally, $A$ pays $B$ one lev for each vertex that is incident to an odd number of edges. What is the maximum amount that $B$ can guarantee to earn?

How many ways are there to arrange the $6$ permutations of the tuple $(1, 2, 3)$ in a sequence, such that each pair of adjacent permutations contains at least one entry in common? For example, a valid such sequence is given by $(3, 2, 1) - (2, 3, 1) - (2, 1, 3) - (1, 2, 3) - (1, 3, 2) - (3, 1, 2)$.

Let $k\leq n$ be two positive integers. IMO-nation has $n$ villages, some of which are connected by a road. For any two villages, the distance between them is the minimum number of toads that one needs to travel from one of the villages to the other, if the traveling is impossible, then the distance is set as infinite. Alice, who just arrived IMO-nation, is doing her quarantine in some place, so she does not know the configuration of roads, but she knows $n$ and $k$. She wants to know whether the furthest two villages have finite distance. To do so, for every phone call she dials to the IMO office, she can choose two villages, and ask the office whether the distance between them is larger than, equal to, or smaller than $k$. The office answers faithfully (infinite distance is larger than $k$). Prove that Alice can know whether the furthest two villages have finite distance between them in at most $2n^2/k$ calls. [i]Proposed by usjl and Cheng-Ying Chang[/i]

[b]p22.[/b] Find the unique ordered pair $(m, n)$ of positive integers such that $x = \sqrt[3]{m} -\sqrt[3]{n}$ satisfies $x^6 + 4x^3 - 36x^2 + 4 = 0$. [b]p23.[/b] Jenny plays with a die by placing it flat on the ground and rolling it along any edge for each step. Initially the face with $1$ pip is face up. How many ways are there to roll the dice for $6$ steps and end with the $1$ face up again? [b]p24.[/b] There exists a unique positive five-digit integer with all odd digits that is divisible by $5^5$. Find this integer. PS. You should use hide for answers.

At the Berkeley Math Tournament, teams are composed of $6$ students, each of whom pick two distinct subject tests out of $5$ choices. How many different distributions across subjects are possible for a team?

Let $n$ and $k$ be positive integers with $1 \leq k \leq n$. A set of cards numbered $1$ to $n$ are arranged randomly in a row from left to right. A person alternates between performing the following moves: [list=a] [*] The leftmost card in the row is moved $k-1$ positions to the right while the cards in positions $2$ through $k$ are each moved one place to the left. [*] The rightmost card in the row is moved $k-1$ positions to the left while the cards in positions $n-k+1$ through $n-1$ are each moved one place to the right. [/list] Determine the probability that after some number of moves the cards end up in order from $1$ to $n$, left to right.

$10$ students bought some books in a bookstore. It is known that every student bought exactly three kinds of books, and any two of them shared at least one kind of book. Determine, with proof, how many students bought the most popular book at least? (Note: the most popular book means most students bought this kind of book)

In a school there are $1200$ students. Each student is part of exactly $k$ clubs. For any $23$ students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of $k$.

Let $n$ be a non-zero natural number, $n \ge 5$. Consider $n$ distinct points in the plane, each colored or white, or black. For each natural $k$ , a move of type $k, 1 \le k <\frac {n} {2}$, means selecting exactly $k$ points and changing their color. Determine the values of $n$ for which, whatever $k$ and regardless of the initial coloring, there is a finite sequence of $k$ type moves, at the end of which all points have the same color.