How many different ways (up to rotation) are there of labeling the faces of a cube with the numbers $1, 2,..., 6$?

To prepare a stay abroad, students meet at a workshop including an excursion. To promote interaction between the students, they are to be distributed to two busses such that not too many of the students in the same bus know each other. Every student knows all those who know her. The number of such pairwise acquaintances is $k$. Prove that it is possible to distribute the students such that the number of pairwise acquaintances in each bus is at most $\frac{k}{3}$.

Take $k\in \mathbb{Z}_{\ge 1}$ and the sets $A_1,A_2,\dots, A_k$ consisting of $x_1,x_2,\dots ,x_k$ positive integers, respectively. For any two sets $A$ and $B$, define $A+B=\{a+b~|~a\in A,~b\in B\}$. Find the least and greatest number of elements the set $A_1+A_2+\dots +A_k$ may have. [i] (Andrei Bâra)[/i]

a) How many distinct ways are there are there of painting the faces of a cube six different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.) b)* How many distinct ways are there are there of painting the faces of a dodecahedron $12$ different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.)

There are $36$ participants at a BdMO event. Some of the participants shook hands with each other. But no two participants shook hands with each other more than once. Each participant recorded the number of handshakes they made. It was found that no two participants with the same number of handshakes made, had shaken hands with each other. Find the maximum possible number of handshakes at the party with proof. (When two participants shake hands with each other, this will be counted as one handshake.)

Maria and Luis play the following game: Maria throws three dice and Luis can select some of them (possibly none) and turn them changing their value for the value in the opposite face of each selected die. Prove that Luis can always play in such a way that the sum of the upper faces of the dice after the change is a multiple of $4$. Note: The game is played with normal dice, that is, the sum of opposite faces is $7$.

There are \(n^2 + n\) numbers, none of which appears more than \(\frac{n^2 + n}{2}\) times. Prove that they can be divided into \((n+1)\) groups of \(n\) numbers each in such a way that the sums of the numbers in these groups are pairwise distinct. [i]Proposed by Anton Trygub[/i]

$n \geq 5$ football teams participate in a round-robin tournament. For every game played, the winner receives 3 points, the loser receives 0 points, and in the event of a draw, both teams receive 1 point. The third-from-bottom team has fewer points than all the teams ranked before it, and more points than the last 2 teams; it won more games than all the teams before it, but fewer games than the 2 teams behind it. Find the smallest possible $n$.

In a hat are $1992$ notes with all numbers from $1$ to $1992$. At random way, two bills are drawn simultaneously from the hat; the difference between the numbers on the two notes are written on a new note, which is placed in the hat, while the two drawn notes thrown away. It continues in this way until there is only one note left in it the hat. Show that there is an even number on this slip of paper.

You wish to color every vertex, edge, face, and the interior of a cube one color each such that no two adjacent objects are the same color. Faces are adjacent if they share an edge. Edges are adjacent if they share a vertex. The interior is adjacent to all of its faces, edges, and vertices. Each face is adjacent to all of its edges and vertices. Each edge is adjacent to both of its vertices. What is the minimum number of colors required to do this?

For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$ $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

p1. A regular $6$-face dice is thrown three times. Calculate the probability that the number of dice points on all three throws is $ 12$? p2. Given two positive real numbers $x$ and $y$ with $xy = 1$. Determine the minimum value of $\frac{1}{x^4}+\frac{1}{4y^4}.$ p3. Known a square network which is continuous and arranged in forming corners as in the following picture. Determine the value of the angle marked with the letter $x$. [img]https://cdn.artofproblemsolving.com/attachments/6/3/aee36501b00c4aaeacd398f184574bd66ac899.png[/img] p4. Find the smallest natural number $n$ such that the sum of the measures of the angles of the $n$-gon, with $n > 6$ is less than $n^2$ degrees. p5. There are a few magic cards. By stating on which card a number is there, without looking at the card at all, someone can precisely guess the number. If the number is on Card $A$ and $B$, then the number in question is $1 + 2$ (sum of corner number top left) cards $A$ and $B$. If the numbers are in $A$, $B$, and $C$, the number what is meant is $1 + 2 + 4$ or equal to $7$ (which is obtained by adding the numbers in the upper left corner of each card $A$,$B$, and $C$). [img]https://cdn.artofproblemsolving.com/attachments/e/5/9e80d4f3bba36a999c819c28c417792fbeff18.png[/img] a. How can this be explained? b. Suppose we are going to make cards containing numbers from $1$ to with $15$ based on the rules above. Try making the cards. [hide=original wording for p5, as the wording isn't that clear]Ada suatu kartu ajaib. Dengan menyebutkan di kartu yang mana suatu bilan gan berada, tanpa melihat kartu sama sekali, seseorang dengan tepat bisa menebak bilangan yang dimaksud. Kalau bilangan tersebut ada pada Kartu A dan B, maka bilangan yang dimaksud adalah 1 + 2 (jumlah bilangan pojok kiri atas) kartu A dan B. Kalau bilangan tersebut ada di A, B, dan C, bilangan yang dimaksud adalah 1 + 2 + 4 atau sama dengan 7 (yang diperoleh dengan menambahkan bilangan-bilangan di pojok kiri atas masing-masing kartu A, B, dan C) a. Bagaimana hal ini bisa dijelaskan? b. Andai kita akan membuat kartu-kartu yang memuat bilangan dari 1 sampai dengan 15 berdasarkan aturan di atas. Coba buatkan kartu-kartunya[/hide]

Is it possible to put pairwise distinct positive integers less than $100$ in the cells of a $4 \times 4$ table so that the products of all the numbers in every column and every row are equal to each other? (N.B. Vasiliev, Moscow))

The Baltic Sea has $2016$ harbours. There are two-way ferry connections between some of them. It is impossible to make a sequence of direct voyages $C_1 - C_2 - ... - C_{1062}$ where all the harbours $C_1, . . . , C_{1062}$ are distinct. Prove that there exist two disjoint sets $A$ and $B$ of $477$ harbours each, such that there is no harbour in $A$ with a direct ferry connection to a harbour in $B.$

A square $ (n \minus{} 1) \times (n \minus{} 1)$ is divided into $ (n \minus{} 1)^2$ unit squares in the usual manner. Each of the $ n^2$ vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemse are regarded as different if at least one vertex is coloured differently in the two schemes.)

From the set of all permutations $f$ of $\{1, 2, ... , n\}$ that satisfy the condition: $f(i) \geq i-1$ $i=1,...,n$ one is chosen uniformly at random. Let $p_n$ be the probability that the chosen permutation $f$ satisfies $f(i) \leq i+1$ $i=1,...,n$ Find all natural numbers $n$ such that $p_n > \frac{1}{3}$.

Find the largest positive constant $C$ such that the following is satisfied: Given $n$ arcs (containing their endpoints) $A_1,A_2,\ldots ,A_n$ on the circumference of a circle, where among all sets of three arcs $(A_i,A_j,A_k)$ $(1\le i< j< k\le n)$, at least half of them has $A_i\cap A_j\cap A_k$ nonempty, then there exists $l>Cn$, such that we can choose $l$ arcs among $A_1,A_2,\ldots ,A_n$, whose intersection is nonempty.

Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$. Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$

In a drawer, there are at most $ 2009$ balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is $ \frac12$. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?

p1. If $x + y + z = 2$, show that $\frac{1}{xy+z-1}+\frac{1}{yz+x-1}+\frac{1}{xz+y-1}=\frac{-1}{(x-1)(y-1)(z-1)}$. p2. Determine the simplest form of $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\frac{5}{3!+4!+5!}+...+\frac{100}{98!+99!+100!}$ p3. It is known that $ABCD$ and $DEFG$ are two parallelograms. Point $E$ lies on $AB$ and point $C$ lie on $FG$. The area of $​​ABCD$ is $20$ units. $H$ is the point on $DG$ so that $EH$ is perpendicular to $DG$. If the length of $DG$ is $5$ units, determine the length of $EH$. [img]https://cdn.artofproblemsolving.com/attachments/b/e/42453bf6768129ed84fbdc81ab7235e780b0e1.png[/img] p4. Each room in the following picture will be painted so that every two rooms which is directly connected to the door is given a different color. If $10$ different colors are provided and $4$ of them can not be used close together for two rooms that are directly connected with a door, determine how many different ways to color the $4$ rooms. [img]https://cdn.artofproblemsolving.com/attachments/4/a/e80a464a282b3fe3cdadde832b2fd38b51a41a.png[/img] 5. The floor of a hall is rectangular $ABCD$ with $AB = 30$ meters and $BC = 15$ meters. A cat is in position $A$. Seeing the cat, the mouse in the midpoint of $AB$ ran and tried to escape from cat. The mouse runs from its place to point $C$ at a speed of $3$ meters/second. The trajectory is a straight line. Watching the mice run away, at the same time from point $A$ the cat is chasing with a speed of $5$ meters/second. If the cat's path is also a straight line and the mouse caught before in $C$, determine an equation that can be used for determine the position and time the mouse was caught by the cat.

Construct a set of $k$ integer weights that allows you to get any total integer weight from $1$ up to $55$ grams even if some of the weights of the set are lost. Consider two versions: (a) $k = 10$, and any one of the weights may be lost; (b) $k = 12$, and any two of the weights may be lost. (D Zvonkin) (In both cases prove that the set found has the property required.)

A corner with arm $n$ is a figure made of $2n-1$ unit squares, such that 2 rectangles $1$ x $(n-1)$ are connected to two adjacent sides of a square $1$ x $1$, so that their unit sides coincide. The squares or a chessboard $100$ x $100$ are colored in 15 colors. We say that a corner with arm 8 is [i]“multicolored”[/i], if it contains each of the colors on the board. What’s the greatest number of corners with arm 8 which could be [i]“mutlticolored”[/i]?

[b]p1.[/b] What is the units digit of $1 + 9 + 9^2 +... + 9^{2015}$ ? [b]p2.[/b] In Fourtown, every person must have a car and therefore a license plate. Every license plate must be a $4$-digit number where each digit is a value between $0$ and $9$ inclusive. However $0000$ is not a valid license plate. What is the minimum population of Fourtown to guarantee that at least two people who have the same license plate? [b]p3.[/b] Two sides of an isosceles triangle $\vartriangle ABC$ have lengths $9$ and $4$. What is the area of $\vartriangle ABC$? [b]p4.[/b] Let $x$ be a real number such that $10^{\frac{1}{x}} = x$. Find $(x^3)^{2x}$. [b]p5.[/b] A Berkeley student and a Stanford student are going to visit each others campus and go back to their own campuses immediately after they arrive by riding bikes. Each of them rides at a constant speed. They first meet at a place $17.5$ miles away from Berkeley, and secondly $10$ miles away from Stanford. How far is Berkeley away from Stanford in miles? [b]p6.[/b] Let $ABCDEF$ be a regular hexagon. Find the number of subsets $S$ of $\{A,B,C,D,E, F\}$ such that every edge of the hexagon has at least one of its endpoints in $S$. [b]p7.[/b] A three digit number is a multiple of $35$ and the sum of its digits is $15$. Find this number. [b]p8.[/b] Thomas, Olga, Ken, and Edward are playing the card game SAND. Each draws a card from a $52$ card deck. What is the probability that each player gets a dierent rank and a different suit from the others? [b]p9.[/b] An isosceles triangle has two vertices at $(1, 4)$ and $(3, 6)$. Find the $x$-coordinate of the third vertex assuming it lies on the $x$-axis. [b]p10.[/b] Find the number of functions from the set $\{1, 2,..., 8\}$ to itself such that $f(f(x)) = x$ for all $1 \le x \le 8$. [b]p11.[/b] The circle has the property that, no matter how it's rotated, the distance between the highest and the lowest point is constant. However, surprisingly, the circle is not the only shape with that property. A Reuleaux Triangle, which also has this constant diameter property, is constructed as follows. First, start with an equilateral triangle. Then, between every pair of vertices of the triangle, draw a circular arc whose center is the $3$rd vertex of the triangle. Find the ratio between the areas of a Reuleaux Triangle and of a circle whose diameters are equal. [b]p12.[/b] Let $a$, $b$, $c$ be positive integers such that gcd $(a, b) = 2$, gcd $(b, c) = 3$, lcm $(a, c) = 42$, and lcm $(a, b) = 30$. Find $abc$. [b]p13.[/b] A point $P$ is inside the square $ABCD$. If $PA = 5$, $PB = 1$, $PD = 7$, then what is $PC$? [b]p14.[/b] Find all positive integers $n$ such that, for every positive integer $x$ relatively prime to $n$, we have that $n$ divides $x^2 - 1$. You may assume that if $n = 2^km$, where $m$ is odd, then $n$ has this property if and only if both $2^k$ and $m$ do. [b]p15.[/b] Given integers $a, b, c$ satisfying $$abc + a + c = 12$$ $$bc + ac = 8$$ $$b - ac = -2,$$ what is the value of $a$? [b]p16.[/b] Two sides of a triangle have lengths $20$ and $30$. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side? [b]p17.[/b] Find the number of non-negative integer solutions $(x, y, z)$ of the equation $$xyz + xy + yz + zx + x + y + z = 2014.$$ [b]p18.[/b] Assume that $A$, $B$, $C$, $D$, $E$, $F$ are equally spaced on a circle of radius $1$, as in the figure below. Find the area of the kite bounded by the lines $EA$, $AC$, $FC$, $BE$. [img]https://cdn.artofproblemsolving.com/attachments/7/7/57e6e1c4ef17f84a7a66a65e2aa2ab9c7fd05d.png[/img] [b]p19.[/b] A positive integer is called cyclic if it is not divisible by the square of any prime, and whenever $p < q$ are primes that divide it, $q$ does not leave a remainder of $1$ when divided by $p$. Compute the number of cyclic numbers less than or equal to $100$. [b]p20.[/b] On an $8\times 8$ chess board, a queen can move horizontally, vertically, and diagonally in any direction for as many squares as she wishes. Find the average (over all $64$ possible positions of the queen) of the number of squares the queen can reach from a particular square (do not count the square she stands on). PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Ten cars are moving at the road. There are some cities at the road. Each car is moving with some constant speed through cities and with some different constant speed outside the cities (different cars may move with different speed). There are 2011 points at the road. Cars don't overtake at the points. Prove that there are 2 points such that cars pass through these points in the same order. [i]S. Berlov[/i]

A [i]form [/i] is the union of squared rectangles whose bases are consecutive unitary segments in a horizontal line that leaves all the rectangles on the same side, and whose heights $m_1, ... , m_n$ satisying $m_1\ge ... \ge m_n$. An [i]angle [/i] in a [i]form [/i] consists of a box $v$ and of all the boxes to the right of $v$ and all the boxes above $v$. The size of a [i]form [/i] of an [i]angle [/i] is the number of boxes it contains. Find the maximum number of [i]angles [/i] of size $11$ in a form of size $400$. [url=http://www.oma.org.ar/enunciados/omr20.htm]source[/url]