Students in a class form groups each of which contains exactly three members such that any two distinct groups have at most one member in common. Prove that, when the class size is $ 46$, there is a set of $ 10$ students in which no group is properly contained.

Let $\{E_1, E_2, \dots, E_m\}$ be a collection of sets such that $E_i \subseteq X = \{1, 2, \dots, 100\}$, $E_i \neq X$, $i = 1, 2, \dots, m$. It is known that every two elements of $X$ is contained together in exactly one $E_i$ for some $i$. Determine the minimum value of $m$.

There are $n$ knights numbered $1$ to $n$ and a round table with $n$ chairs. The first knight chooses his chair, and from him, the knight number $k+1$ sits $ k$ places to the right of knight number $k$ , for all $1 \le k\le n-1$ (occupied and empty seats are counted). In particular, the second knight sits next to the first. Find all values ​​of $n$ such that the $n$ gentlemen occupy the $n$ chairs following the described procedure.

In Happy City there are $2014$ citizens called $A_1, A_2, \dots , A_{2014}$. Each of them is either [i]happy[/i] or [i]unhappy[/i] at any moment in time. The mood of any citizen $A$ changes (from being unhappy to being happy or vice versa) if and only if some other happy citizen smiles at $A$. On Monday morning there were $N$ happy citizens in the city. The following happened on Monday during the day: the citizen $A_1$ smiled at citizen $A_2$, then $A_2$ smiled at $A_3$, etc., and, finally, $A_{2013}$ smiled at $A_{2014}$. Nobody smiled at anyone else apart from this. Exactly the same repeated on Tuesday, Wednesday and Thursday. There were exactly $2000$ happy citizens on Thursday evening. Determine the largest possible value of $N$.

There are 10 stones of different weights with distinct pairwise sums. We have a special two-tiered balance scale such that only two stones can be put on each cup and then we understand which cup is heavier. Prove that having this scale you can either find the heaviest or the lightest stone.

Albert and Betty are playing the following game. There are $100$ blue balls in a red bowl and $100$ red balls in a blue bowl. In each turn a player must make one of the following moves: a) Take two red balls from the blue bowl and put them in the red bowl. b) Take two blue balls from the red bowl and put them in the blue bowl. c) Take two balls of different colors from one bowl and throw the balls away. They take alternate turns and Albert starts. The player who first takes the last red ball from the blue bowl or the last blue ball from the red bowl wins. Determine who has a winning strategy.

Let $S$ be the set of all $5^6$ positive integers, whose decimal representation consists of exactly 6 odd digits. Find the number of solutions $(x,y,z)$ of the equation $x+y=10z$, where $x\in S$, $y\in S$, $z\in S$.

We say that a set $\displaystyle A$ of non-zero vectors from the plane has the property $\displaystyle \left( \mathcal S \right)$ iff it has at least three elements and for all $\displaystyle \overrightarrow u \in A$ there are $\displaystyle \overrightarrow v, \overrightarrow w \in A$ such that $\displaystyle \overrightarrow v \neq \overrightarrow w$ and $\displaystyle \overrightarrow u = \overrightarrow v + \overrightarrow w$. (a) Prove that for all $\displaystyle n \geq 6$ there is a set of $\displaystyle n$ non-zero vectors, which has the property $\displaystyle \left( \mathcal S \right)$. (b) Prove that every finite set of non-zero vectors, which has the property $\displaystyle \left( \mathcal S \right)$, has at least $\displaystyle 6$ elements. [i]Mihai Baluna[/i]

Three families of parallel lines are drawn,$10$ lines each, are drawn. What is the greatest number of triangles they can cut from plane?

Each of eight boxes contains six balls. Each ball has been colored with one of $n$ colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justification, the smallest integer $n$ for which this is possible.

[u]Round 1[/u] [b]p1.[/b] Compute $(1 - 2(3 - 4(5 - 6)))(7 - (8 - 9))$. [b]p2.[/b] How many numbers are in the set $\{20, 21, 22, ..., 88, 89\}$? [b]p3.[/b] Three times the complement of the supplement of an angle is equal to $60$ degrees less than the angle itself. Find the measure of the angle in degrees. [u]Round 2[/u] [b]p4.[/b] A positive number is decreased by $10\%$, then decreased by $20\%$, and finally increased by $30\%$. By what percent has this number changed from the original? Give a positive answer for a percent increase and a negative answer for a percent decrease. [b]p5.[/b] What is the area of the triangle with vertices at $(2, 3)$, $(8, 11)$, and $(13, 3)$? [b]p6.[/b] There are three bins, each containing red, green, and/or blue pens. The first bin has $0$ red, $0$ green, and $3$ blue pens, the second bin has $0$ red, $2$ green, and $4$ blue pens, and the final bin has $1$ red, $5$ green, and $6$ blue pens. What is the probability that if one pen is drawn from each bin at random, one of each color pen will be drawn? [u]Round 3[/u] [b]p7.[/b] If a and b are positive integers and $a^2 - b^2 = 23$, what is the value of $a$? [b]p8.[/b] Find the prime factorization of the greatest common divisor of $2^3\cdot 3^2\cdot 5^5\cdot 7^4$ and $2^4\cdot 3^1\cdot 5^2\cdot 7^6$. [b]p9.[/b] Given that $$a + 2b + 3c = 5$$ $$2a + 3b + c = -2$$ $$3a + b + 2c = 3,$$ find $3a + 3b + 3c$. [u]Round 4[/u] [b]p10.[/b] How many positive integer divisors does $11^{20}$ have? [b]p11.[/b] Let $\alpha$ be the answer to problem $10$. Find the real value of $x$ such that $2^{x-5} = 64^{x/\alpha}$. [b]p12.[/b] Let $\beta$ be the answer to problem $11$. Triangle $LMT$ has a right angle at $M$, $LM = \beta$, and $LT = 4\beta - 3$. If $Z$ is the midpoint of $LT$, what is the length$ MZ$? PS. You should use hide for answers. Rounds 5-8 are [url=https://artofproblemsolving.com/community/c3h3133709p28395558]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134133p28400917]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $T$ be a triangulation of a $100$-gon.We construct $P(T)$ by copying the same $100$-gon and drawing a diagonal if it was not drawn in $T$ an there is a quadrilateral with this diagonal and two other vertices so that all the sides and diagonals(Except the one we are going to draw) are present in $T$.Let $f(T)$ be the number of intersections of diagonals in $P(T)$.Find the minimum and maximum of $f(T)$.

Consider a rectangle whose lengths of sides are natural numbers. If someone places as many squares as possible, each with area $3$, inside of the given rectangle, such that the sides of the squares are parallel to the rectangle sides, then the maximal number of these squares fill exactly half of the area of the rectangle. Determine the dimensions of all rectangles with this property.

Thirty rays with the origin at the same point are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$. (E. Barabanov)

A meeting was attended by $n$ people. They are welcome to occupy the $k$ table provided $\left( k \le \frac{n}{2} \right)$. Each table is occupied by at least two people. When the meeting begins, the moderator selects two people from each table as representatives for talk to. Suppose that $A$ is the number of ways to choose representatives to speak. Determine the maximum value of $A$ that is possible.

Let $n$ be a positive integer. There is a collection of cards that meets the following properties: $\bullet$Each card has a number written in the form $m!$, where $m$ is a positive integer. $\bullet$For every positive integer $t\le n!$, it is possible to choose one or more cards from the collection in such a way $\text{ }$that the sum of the numbers of those cards is $t$. Determine, based on $n$, the smallest number of cards that this collection can have.

Let $n$ be a positive integer. Morgane has coloured the integers $1,2,\ldots,n$. Each of them is coloured in exactly one colour. It turned out that for all positive integers $a$ and $b$ such that $a<b$ and $a+b \leqslant n$, at least two of the integers among $a$, $b$ and $a+b$ are of the same colour. Prove that there exists a colour that has been used for at least $2n/5$ integers. \\ \\ (Vincent Jugé)

There are $n$ students in a class, and some pairs of these students are friends. Among any six students, there are two of them that are not friends, and for any pair of students that are not friends there is a student among the remaining four that is friends with both of them. Find the maximum value of $n$.

In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.

For a set $S$ of at least $3$ points in the plane, let $d_{\text{min}}$ denote the minimal distance between two different points in $S$ and $d_{\text{max}}$ the maximal distance between two different points in $S$. For a real $c>0$, a set $S$ will be called $c$-[i]balanced[/i] if \[\frac{d_{\text{max}}}{d_{\text{min}}}\leq c|S|\] Prove that there exists a real $c>0$ so that for every $c$-balanced set of points $S$, there exists a triangle with vertices in $S$ that contains at least $\sqrt{|S|}$ elements of $S$ in its interior or on its boundary.

Natural numbers are placed in an infinite grid. Such that the number in each cell is equal to the number of its adjacent cells having the same number. Find the most distinct numbers this infinite grid can have. (Two cells of the grid are adjacent if they have a common vertex)

Sixteen squares of an $8\times 8$ chessboard are chosen so that there are exactly lwo in each row and two in each column. Prove that eight white pawns and eight black pawns can be placed on these sixteen squares so that there is one white pawn and one black pawn in each row and in cach colunm.

Let $n\ge 2$ be a positive integer. The integers $1,2,\cdots,n$ are written on a board. In a move, Alice can pick two integers written on the board $a\neq b$ such that $a+b$ is an even number, erase both $a$ and $b$ from the board and write the number $\frac{a+b}{2}$ on the board instead. Find all $n$ for which Alice can make a sequence of moves so that she ends up with only one number remaining on the board. [b]Note.[/b] When $n=3$, Alice changes $(1,2,3)$ to $(2,2)$ and can't make any further moves. [i]Proposed by Rohan Goyal[/i]

Let $t(A)$ denote the sum of elements of a nonempty set $A$ of integers, and define $t(\emptyset)=0$. Find a set $X$ of positive integers such that for every integers $k$ there is a unique ordered pair of disjoint subsets $(A_{k},B_{k})$ of $X$ such that $t(A_{k})-t(B_{k}) = k$.

At a two-round volleyball tournament participated 99 teams. Each played a match at home and a match away. Each team won exactly half of their home matches and exactly half of their away matches. Prove that one of the teams beat another team twice. [i]Proposed by M. Antipov[/i]