There are infinitely many boxes - initially one of them contains $n$ balls and all others are empty. On a single move we take some balls from a non-empty box and put them into an empty box and on a sheet of paper we write down the product of the resulting amount of balls in the two boxes. After some moves, the sum of all numbers on the sheet of paper became $2023$. What is the smallest possible value of $n$?

Let $n$ be a positive integer. Find the smallest positive integer $k$ such that for any set $S$ of $n$ points in the interior of the unit square, there exists a set of $k$ rectangles such that the following hold: [list=disc] [*]The sides of each rectangle are parallel to the sides of the unit square. [*]Each point in $S$ is [i]not[/i] in the interior of any rectangle. [*]Each point in the interior of the unit square but [i]not[/i] in $S$ is in the interior of at least one of the $k$ rectangles [/list] (The interior of a polygon does not contain its boundary.) [i]Holden Mui[/i]

We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black. Consider a finite graph with all vertices colored white. Now, we can do the following operation: Switch a vertex and simultaneously switch all of its neighbours (i. e. all vertices connected to this vertex by an edge). Can we, just by performing this operation several times, obtain a graph with all vertices colored black? [It is assumed that our graph has no loops (a [i]loop[/i] means an edge connecting one vertex with itself) and no multiple edges (a [i]multiple edge[/i] means a pair of vertices connected by more than one edge).]

A football league has $n$ teams. Each team plays one game with every other team. Each win is awarded $2$ points, each tie $1$ point, and each loss $0$ points. After the league is over, the following statement is true: for every subset $S$ of teams in the league, there is a team (which may or may not be in $S$) such that the total points the team obtained by playing all the teams in $S$ is odd. Prove that $n$ is even.

Consider the segment $[0; 1]$. At each step we may split one of the available segments into two new segments and write the product of lengths of these two new segments onto a blackboard. Prove that the sum of the numbers on the blackboard never will exceed $1/2$. [i]Mikhail Lukin[/i]

How many [i]connected subsequences [/i](i.e, consisting of one element or consecutive elements) of the following sequence are there: $1,2,...,100$? [b]A.[/b] $1010$ [b]B.[/b] $2020$ [b]C.[/b] $3030$ [b]D.[/b] $4040$ [b]E.[/b] $5050$

Find the greatest $n$ for which there exist $n$ lines in space, passing through a single point, such that any two of them form the same angle.

Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8,4$ and let $n = (10a + b)^{10c+d}$. Find the probability that $1984!$ is divisible by $n.$

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]

Each of the 450 members of a parliament gives a slap in the face to exactly one of his colleagues. Prove that after that they can choose a committee consisting of 150 members, none of whom has been slapped in the face by any other member of the committee. (Folklore)

Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$, $x$ divides at most one other element in $A$. Prove that \[\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil. \]

Consider a set $S$ of $200$ points on the plane such that $100$ points are the vertices of a convex polygon $A$ and the other $100$ points are in the interior of the polygon. Moreover, no three of the given points are collinear. A triangulation is a way to partition the interior of the polygon $A$ into triangles by drawing the edges between some two points of S such that any two edges do not intersect in the interior, and each point in $S$ is the vertex of at least one triangle. 1. Prove that the number of edges does not depend on the triangulation. 2. Show that for any triangulation, one can color each triangle by one of three given colors such that any two adjacent triangles have different colors.

Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1,2, ..., 2015$.

Consider a convex $15$- gon with perimeter $21$. Show that there one can select three distinct pairs of vertices that form a triangle with area less than $1$. [hide=original wording of second sentence]Zeige, dass man davon drei paarweise verschiedene Eckpunkte auswählen kann, die ein Dreieck mit Fläche kleiner als 1 bilden.[/hide]

a) $101$ numbers are selected from the set $1, 2, . . . , 200$. Prove that among the numbers selected there is a pair in which one number is divisible by the other. b) One number less than $16$, and $99$ other numbers are selected from the set $1, 2, . . . , 200$. Prove that among the selected numbers there are two such that one divides the other.

The harmonic table is a triangular array: $1$ $\frac 12 \qquad \frac 12$ $\frac 13 \qquad \frac 16 \qquad \frac 13$ $\frac 14 \qquad \frac 1{12} \qquad \frac 1{12} \qquad \frac 14$ Where $a_{n,1} = \frac 1n$ and $a_{n,k+1} = a_{n-1,k} - a_{n,k}$ for $1 \leq k \leq n-1.$ Find the harmonic mean of the $1985^{th}$ row.

Red, blue, and green children are arranged in a circle. When a teacher asked the red children that have a green neighbor to raise their hands, $20$ children raised their hands. When she asked the blue children that have a green neighbor to raise their hands, $25$ children raised their hands. Prove that some child that raised her hand had two green neighbors.

White and black checkers are put on the squares of an $n\times n$ chessboard $(n\ge2)$ according to the following rule. Initially, a black checker is put on an arbitrary square. In every consequent step, a white checker is put on a free square, whereby all checkers on the squares neighboring by side are replaced by checkers of the opposite colors. This process is continued until there is a checker on every square. Prove that in the final configuration there is at least one black checker.

[u]Round 5[/u] [b]p13.[/b] Express the number $3024_8$ in base $2$. [b]p14.[/b] $\vartriangle ABC$ has a perimeter of $10$ and has $AB = 3$ and $\angle C$ has a measure of $60^o$. What is the maximum area of the triangle? [b]p15.[/b] A weighted coin comes up as heads $30\%$ of the time and tails $70\%$ of the time. If I flip the coin $25$ times, howmany tails am I expected to flip? [u]Round 6[/u] [b]p16.[/b] A rectangular box with side lengths $7$, $11$, and $13$ is lined with reflective mirrors, and has edges aligned with the coordinate axes. A laser is shot from a corner of the box in the direction of the line $x = y = z$. Find the distance traveled by the laser before hitting a corner of the box. [b]p17.[/b] The largest solution to $x^2 + \frac{49}{x^2}= 2018$ can be represented in the form $\sqrt{a}+\sqrt{b}$. Compute $a +b$. [b]p18.[/b] What is the expected number of black cards between the two jokers of a $54$ card deck? [u]Round 7[/u] p19. Compute ${6 \choose 0} \cdot 2^0 + {6 \choose 1} \cdot 2^1+ {6 \choose 2} \cdot 2^2+ ...+ {6 \choose 6} \cdot 2^6$. [b]p20.[/b] Define a sequence by $a_1 =5$, $a_{n+1} = a_n + 4 * n -1$ for $n\ge 1$. What is the value of $a_{1000}$? [b]p21.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle B = 15^o$ and $\angle C = 30^o$. Let $D$ be the point such that $\vartriangle ADC$ is an isosceles right triangle where $D$ is in the opposite side from $A$ respect to $BC$ and $\angle DAC = 90^o$. Find the $\angle ADB$. [u]Round 8[/u] [b]p22.[/b] Say the answer to problem $24$ is $z$. Compute $gcd (z,7z +24).$ [b]p23.[/b] Say the answer to problem $22$ is $x$. If $x$ is $1$, write down $1$ for this question. Otherwise, compute $$\sum^{\infty}_{k=1} \frac{1}{x^k}$$ [b]p24.[/b] Say the answer to problem $23$ is $y$. Compute $$\left \lfloor \frac{y^2 +1}{y} \right \rfloor$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165983p28809209]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166045p28809814]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A rook starts moving on an infinite chessboard, alternating horizontal and vertical moves. The length of the first move is one square, of the second – two squares, of the third – three squares and so on. a) Is it possible for the rook to arrive at its starting point after exactly $2013$ moves? b) Find all $n$ for which it possible for the rook to come back to its starting point after exactly $n$ moves.

Prove that it is possible to color each positive integers with one of three colors so that the following conditions are satisfied: $i)$ For each $n\in\mathbb{N}_{0}$ all positive integers $x$ such that $2^n\le x<2^{n+1}$ have the same color. $ii)$ There are no positive integers $x,y,z$ of the same color (except $x=y=z=2$) such that $x+y=z^2.$

Given is a positive integer $k$. There are $n$ points chosen on a line, such the distance between any two adjacent points is the same. The points are colored in $k$ colors. For each pair of monochromatic points such that there are no points of the same color between them, we record the distance between these two points. If all distances are distinct, find the largest possible $n$.

Points with integer coordinates in cartesian space are called lattice points. We color $2000$ lattice points blue and $2000$ other lattice points red in such a way that no two blue-red segments have a common interior point (a segment is blue-red if its two endpoints are colored blue and red). Consider the smallest rectangular parallelepiped that covers all the colored points. (a) Prove that this rectangular parallelepiped covers at least $500,000$ lattice points. (b) Give an example of a coloring for which the considered rectangular paralellepiped covers at most $8,000,000$ lattice points.

Let $G$ is a graph and $x$ is a vertex of $G$. Define the transformation $\varphi_{x}$ over $G$ as deleting all incident edges with respect of $x$ and drawing the edges $xy$ such that $y\in G$ and $y$ is not connected with $x$ with edge in the beginning of the transformation. A graph $H$ is called $G-$[i]attainable[/i] if there exists a sequece of such transformations which transforms $G$ in $H.$ Let $n\in\mathbb{N}$ and $4|n.$ Prove that for each graph $G$ with $4n$ vertices and $n$ edges there exists $G-$[i]attainable[/i] graph with at least $9n^{2}/4$ triangles.

We consider a square board of $1000\times 1000$ with $1000000$ squares $1\times 1$ . A piece placed on a square [i]threatens[/i] all squares on the board that are inside a $19\times 19$ square. with a center in the square where the piece is placed, and with sides parallel to those of the board, except for the squares in the same row and those in the same column. Determine the maximum number of pieces that can be placed on the board so that no two pieces threaten each other.