There are $ 12$ students in a classroom; $6$ of them are Democrats and 6 of them are Republicans. Every hour the students are randomly separated into four groups of three for political debates. If a group contains students from both parties, the minority in the group will change his/her political alignment to that of the majority at the end of the debate. What is the expected amount of time needed for all $ 12$ students to have the same political alignment, in hours?

Find all possible values of $n \ge 1$ for which there exist $n$ consecutive positive integers whose sum is a prime number.

Each face of a cube is divided into $n^2$ equal squares. The vertices of the squares are called [i]nodes[/i], so each face has $(n+1)^2$ nodes. $(a)$ If $n=2$,does there exist a closed polygonal line whose links are sids of the squares and which passes through each node exactly once? $(b)$ Prove that, for each $n$, such a polygonal line divides the surface area of the cube into two equal parts

There are $2n-1$ twoelement subsets of set $1,2,...,n$. Prove that one can choose $n$ out of these such that their union contains no more than $\frac{2}{3}n+1$ elements.

We colour every square of a $4\times 19$ chess board with one of the colours red, green and blue. Prove that however this colouring is done, we can always find two horizontal rows and two vertical columns such that the $4$ squares on the intersections of these lines all have the same colour.

In any rectangular game board with black and white squares, call a row $X$ a mix of rows $Y$ and $Z$ whenever each cell in row $X$ has the same colour as either the cell of the same column in row $Y$ or the cell of the same column in row $Z$. Let a natural number $m \ge 3$ be given. In some rectangular board, black and white squares lie in such a way that all the following conditions hold. 1) Among every three rows of the board, one is a mix of two others. 2) For every two rows of the board, their corresponding cells in at least one column have different colours. 3) For every two rows of the board, their corresponding cells in at least one column have equal colours. 4) It is impossible to add a new row with each cell either black or white to the board in a way leaving both conditions 1) and 2) still in force Find all possibilities of what can be the number of rows of the board.

Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card. Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$. [i]Proposed by Carl Schildkraut and Colin Tang[/i]

100 couples are invited to a traditional Modolvan dance. The $200$ people stand in a line, and then in a $\textit{step}$, (not necessarily adjacent) many swap positions. Find the least $C$ such that whatever the initial order, they can arrive at an ordering where everyone is dancing next to their partner in at most $C$ steps.

Let $n$ be a positive integer and $A$ be a family of subsets of the set $\{1,2,...,n\},$ none of which contains another subset from A . Find the largest possible cardinality of $A$ .

A finite set of [i]axis parallel [/i]cubes in space has the property of each point of the room is located in a maximum of M different cubes. Show that you can divide the amount of cubes in $8 (M - 1) + 1$ subsets (or less) with the property that the cubes in each subset lacks common points. (An axis parallel cube is a cube whose edges are parallel to the coordinate axes.)

I have a $2$ by $4$ grid of squares; how many ways can I shade at least one of the squares such that no two shaded squares share an edge?

p1. From the measurement of the height of nine trees obtained data as following. a) There are three different measurement results (in meters) b) All data are positive numbers c) Mean$ =$ median $=$ mode $= 3$ d) The sum of the squares of all data is $87.$ Determine all possible heights of the nine trees. p2. If $x$ and $y$ are integers, find the number of pairs $(x,y)$ that satisfy $|x|+|y|\le 50$. p3. The plane figure $ABCD$ on the side is a trapezoid with $AB$ parallel to $CD$. Points $E$ and $F$ lie on $CD$ so that $AD$ is parallel to $BE$ and $AF$ is parallel to $BC$. Point $H$ is the intersection of $AF$ with $BE$ and point $G$ is the intersection of $AC$ with $BE$. If the length of $AB$ is $4$ cm and the length of $CD$ is $10$ cm, calculate the ratio of the area of ​​the triangle $AGH$ to the area of ​​the trapezoid $ABCD$. [img]https://cdn.artofproblemsolving.com/attachments/c/7/e751fa791bce62f091024932c73672a518a240.png[/img] p4. A prospective doctor is required to intern in a hospital for five days in July $2011$. The hospital leadership gave the following rules: a) Internships may not be conducted on two consecutive days. b) The fifth day of internship can only be done after four days counted since the fourth day of internship. Suppose the fourth day of internship is the date $20$, then the fifth day of internship can only be carried out at least the date $24$. Determine the many possible schedule options for the prospective doctor. p5. Consider the following sequences of natural numbers: $5$, $55$, $555$, $5555$, $55555$, $...$ ,$\underbrace{\hbox{5555...555555...}}_{\hbox{n\,\,numbers}}$ . The above sequence has a rule: the $n$th term consists of $n$ numbers (digits) $5$. Show that any of the terms of the sequence is divisible by $2011$.

In a blackboard there are $K$ circles in a row such that one of the numbers $1,...,K$ is assigned to each circle from the left to the right. Change of situation of a circle is to write in it or erase the number which is assigned to it.At the beginning no number is written in its own circle. For every positive divisor $d$ of $K$ ,$1\leq d\leq K$ we change the situation of the circles in which their assigned numbers are divisible by $d$,performing for each divisor $d$ $K$ changes of situation. Determine the value of $K$ for which the following holds;when this procedure is applied once for all positive divisors of $K$ ,then all numbers $1,2,3,...,K$ are written in the circles they were assigned in.

In how many ways can we color exactly $k$ vertices of an $n$-gon in red such that any $2$ consecutive vertices are not both red. (Vertices are considered to be labeled)

In a convex $2008$-gon some of the diagonals are coloured red and the rest blue, so that every vertex is an endpoint of a red diagonal and no three red diagonals concur at a point. It's known that every blue diagonal is intersected by a red diagonal in an interior point. Find the minimal number of intersections of red diagonals.

[b]p25.[/b] You are given that $1000!$ has $2568$ decimal digits. Call a permutation $\pi$ of length $1000$ good if $\pi(2i) > \pi (2i - 1)$ for all $1 \le i \le 500$ and $\pi (2i) > \pi (2i + 1)$ for all $1 \le i \le 499$. Let $N$ be the number of good permutations. Estimate $D$, the number of decimal digits in $N$. You will get $\max \left( 0, 25 - \left\lceil \frac{|D-X|}{10} \right\rceil \right)$ points, where $X$ is the true answer. [b]p26.[/b] A year is said to be [i]interesting [/i] if it is the product of $3$, not necessarily distinct, primes (for example $2^2 \cdot 5$ is interesting, but $2^2 \cdot 3 \cdot 5$ is not). How many interesting years are there between $ 5000$ and $10000$, inclusive? For an estimate of $E$, you will get $\max \left( 0, 25 - \left\lceil \frac{|E-X|}{10} \right\rceil \right)$ points, where $X$ is the true answer. [b]p27.[/b] Sam chooses $1000$ random lattice points $(x, y)$ with $1 \le x, y \le 1000$ such that all pairs $(x, y)$ are distinct. Let $N$ be the expected size of the maximum collinear set among them. Estimate $\lfloor 100N \rfloor$. Let $S$ be the answer you provide and $X$ be the true value of $\lfloor 100N \rfloor$. You will get $\max \left( 0, 25 - \left\lceil \frac{|S-X|}{10} \right\rceil \right)$ points for your estimate. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $m$ and $n$ be natural numbers. Every one of the $m*n$ squares of the $m*n$ board is colored either black or white, so that no 2 neighbouring squares are the same color(the board is colored like in chess").In one step we can pick 2 neighbouring squares and change their colors like this: [b]- [/b]a white square becomes black; [b]-[/b]a black square becomes blue; [b]-[/b]a blue square becomes white. For which $m$ and $n$ can we ,in a finite sequence of these steps, switch the starting colors from white to black and vice versa.

Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$, there exist 3 numbers $a$, $b$, $c$ among them satisfying $abc=m$.

Let $K=(V,E)$ be a finite, simple, complete graph. Let $d$ be a positive integer. Let $\phi:E\to \mathbb{R}^d$ be a map from the edge set to Euclidean space, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$, and the points assigned to the edges of any triangle in $K$ are collinear. Show that the range of $\phi$ is contained in a line.

A merida path of order $2n$ is a lattice path in the first quadrant of $xy$- plane joining $(0,0)$ to $(2n,0)$ using three kinds of steps $U=(1,1)$, $D= (1,-1)$ and $L= (2,0)$, i.e. $U$ joins $x,y)$ to $(x+1,y+1)$ etc... An ascent in a merida path is a maximal string of consecutive steps of the form $U$. If $S(n,k)$ denotes the number of merdia paths of order $2n$ with exactly $k$ ascents, compute $S(n,1)$ and $S(n,n-1)$.

A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset $S$ of dominoes is chosen. Is it true that at least one of the following 2 statements is false? (1) There are $2022$ more horizontal dominoes than vertical dominoes in $S$. (2) The cells covered by the dominoes in $S$ can be tiled completely and exactly by $L$-shaped tetrominoes.

Consider $2018$ pairwise crossing circles no three of which are concurrent. These circles subdivide the plane into regions bounded by circular $edges$ that meet at $vertices$. Notice that there are an even number of vertices on each circle. Given the circle, alternately colour the vertices on that circle red and blue. In doing so for each circle, every vertex is coloured twice- once for each of the two circle that cross at that point. If the two colours agree at a vertex, then it is assigned that colour; otherwise, it becomes yellow. Show that, if some circle contains at least $2061$ yellow points, then the vertices of some region are all yellow. Proposed by [i]India[/i]

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

Let \(a\), \(b\), and \(n\) be positive integers. A lemonade stand owns \(n\) cups, all of which are initially empty. The lemonade stand has a [i]filling machine[/i] and an [i]emptying machine[/i], which operate according to the following rules: [list] [*]If at any moment, \(a\) completely empty cups are available, the filling machine spends the next \(a\) minutes filling those \(a\) cups simultaneously and doing nothing else. [*]If at any moment, \(b\) completely full cups are available, the emptying machine spends the next \(b\) minutes emptying those \(b\) cups simultaneously and doing nothing else. [/list] Suppose that after a sufficiently long time has passed, both the filling machine and emptying machine work without pausing. Find, in terms of \(a\) and \(b\), the least possible value of \(n\). [i]Proposed by Raymond Feng[/i]

Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$, parallel with $d,$ but distinct from $d$, which also contains at least two points from $M$.