There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.

Let $ n $ and $ r $ be positive integers and $ A $ be a set of lattice points in the plane such that any open disc of radius $ r $ contains a point of $ A $. Show that for any coloring of the points of $ A $ in $ n $ colors there exists four points of the same color which are the vertices of a rectangle.

On a $1\times n$ board there are $n-1$ separating edges between neighbouring cells. Initially, none of the edges contain matches. During a move of size $0 < k < n$ a player chooses a $1\times k$ sub-board which contains no matches inside, and places a matchstick on all of the separating edges bordering the sub-board that don’t already have one. A move is considered legal if at least one matchstick can be placed and if either $k = 1$ or $k{}$ is divisible by 4. Two players take turns making moves, the player in turn must choose one of the available legal moves of the largest size $0 < k < n$ and play it. If someone does not have a legal move, the game ends and that player loses. [i]Beat the organisers twice in a row in this game! First the organisers determine the value of $n{}$, then you get to choose whether you want to play as the first or the second player.[/i]

Given a $15\times 15$ chessboard. We draw a closed broken line without self-intersections such that every edge of the broken line is a segment joining the centers of two adjacent cells of the chessboard. If this broken line is symmetric with respect to a diagonal of the chessboard, then show that the length of the broken line is $\leq 200$.

There are $2010$ cities in country, and every two are connected by road. Businessman and Road Ministry play next game. Every morning Businessman buys one road and every evening Ministry destroys 10 free roads. Can Business create cyclic route without self-intersections through exactly $11$ different cities?

The nation of CMIMCland consists of 8 islands, none of which are connected. Each citizen wants to visit the other islands, so the government will build bridges between the islands. However, each island has a volcano that could erupt at any time, destroying that island and any bridges connected to it. The government wants to guarantee that after any eruption, a citizen from any of the remaining $7$ islands can go on a tour, visiting each of the remaining islands exactly once and returning to their home island (only at the end of the tour). What is the minimum number of bridges needed?

[b]p1.[/b] What is the largest distance between any two points on a regular hexagon with a side length of one? [b]p2.[/b] For how many integers $n \ge 1$ is $\frac{10^n - 1}{9}$ the square of an integer? [b]p3.[/b] A vector in $3D$ space that in standard position in the first octant makes an angle of $\frac{\pi}{3}$ with the $x$ axis and $\frac{\pi}{4}$ with the $y$ axis. What angle does it make with the $z$ axis? [b]p4.[/b] Compute $\sqrt{2012^2 + 2012^2 \cdot 2013^2 + 2013^2} - 2012^2$. [b]p5.[/b] Round $\log_2 \left(\sum^{32}_{k=0} {{32} \choose k} \cdot 3^k \cdot 5^k\right)$ to the nearest integer. [b]p6.[/b] Let $P$ be a point inside a ball. Consider three mutually perpendicular planes through $P$. These planes intersect the ball along three disks. If the radius of the ball is $2$ and $1/2$ is the distance between the center of the ball and $P$, compute the sum of the areas of the three disks of intersection. [b]p7.[/b] Find the sum of the absolute values of the real roots of the equation $x^4 - 4x - 1 = 0$. [b]p8.[/b] The numbers $1, 2, 3, ..., 2013$ are written on a board. A student erases three numbers $a, b, c$ and instead writes the number $$\frac12 (a + b + c)\left((a - b)^2 + (b - c)^2 + (c - a)^2\right).$$ She repeats this process until there is only one number left on the board. List all possible values of the remainder when the last number is divided by 3. [b]p9.[/b] How many ordered triples of integers $(a, b, c)$, where $1 \le a, b, c \le 10$, are such that for every natural number $n$, the equation $(a + n)x^2 + (b + 2n)x + c + n = 0$ has at least one real root? Problems' source (as mentioned on official site) is Gator Mathematics Competition. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A graph $G$ with $n$ vertices is called [i]cool[/i] if we can label each vertex with a different positive integer not greater than $\frac{n^2}{4}$ and find a set of non-negative integers $D$ so that there is an edge between two vertices iff the difference between their labels is in $D$. Show that if $n$ is sufficiently large we can always find a graph with $n$ vertices which is not cool.

$16$ people sit around a circular table. After some time, they all stand up and sit down in either the chair they were previously sitting on or on a chair next to it. Determine the number of ways that this can be done. Note: two or more people cannot sit on the same chair.

Some time ago there was a war across the world. In the plane $n$ lines are moving, with the regions contained by the lines being the territories of the countries at war. Each line moves parallel to itself with constant speed (each with its own speed), and no line can reverse its direction. Some of the original countries disappeared (a country disappears iff its area is converted to zero) and within the course of the time, other countries appeared. After some time, the presidents of the existing countries made a treaty to end the war, created the United Nations, and all borders ceased movement. The UN then counted the total numbers of sovereign states that were destroyed and the existing ones, obtaining a total of $k$. Prove that $k\leq \frac{n^3+5n}{6}+1$. Is is possible to have equality?

There is a $m \times (m-1)$ board. (i.e. there are $m+1$ horizontal lines and $m$ vertical lines) A stone is put on an intersection of the lowest horizontal line. Now two players move this stone with the following rules. (i) Each players move the stone to a neighboring intersection along a segment, by turns. (ii) A segment, which is already passed by the stone, cannot be used more. (iii) One who cannot move the stone anymore loses. Prove that there is a winning strategy for the former player.

Let $\mathcal S$ be a set of $16$ points in the plane, no three collinear. Let $\chi(S)$ denote the number of ways to draw $8$ lines with endpoints in $\mathcal S$, such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of $\chi(\mathcal S)$ across all such $\mathcal S$. [i]Ankan Bhattacharya[/i]

Sixteen people for groups of four people such that each two groups have at most two members in common. Prove that there exists a set of six people in which every group is not properly contained in it.

Ivan is playing Lego with $4n^2$ $1 \times 2$ blocks. First, he places $2n^2$ $1 \times 2$ blocks to fit a $2n \times 2n$ square as the bottom layer. Then he builds the top layer on top of the bottom layer using the remaining $2n^2$ $1 \times 2$ blocks. Note that the blocks in the bottom layer are connected to the blocks above it in the top layer, just like real Lego blocks. He wants the whole two-layered building to be connected and not in seperate pieces. Prove that if he can do so, then the four $1\times 2$ blocks connecting the four corners of the bottom layer, must be all placed horizontally or all vertically. [i]Proposed by Ivan Chan Kai Chin[/i]

There are $15$ boys and $15$ girls in the class. The first girl is friends with $4$ boys, the second with $5$, the third with $6$, . . . , the $11$th with $14$, and each of the other four girls is friends with all the boys. It turned out that there are exactly $3 \cdot 2^{25}$ ways to split the entire class into pairs, so that each pair has a boy and a girl who are friends. Prove that any of the friends of the first girl are friends with all the other girls too. [i]G.M.Sharafetdinova[/i]

Maisy is at the origin of the coordinate plane. On her first step, she moves $1$ unit up. On her second step, she moves $ 1$ unit to the right. On her third step, she moves $2$ units up. On her fourth step, she moves $2$ units to the right. She repeats this pattern with each odd-numbered step being $ 1$ unit more than the previous step. Given that the point that Maisy lands on after her $21$st step can be written in the form $(x, y)$, find the value of $x + y$. Proposed by Audrey Chun

(a) Let be given $2n$ distinct points on a circumference, $n$ of which are red and $n$ are blue. Prove that one can join these points pairwise by $n$ segments so that no two segments intersect and the endpoints of each segments have different colors. (b) Show that the statement from (a) remains valid if the points are in an arbitrary position in the plane so that no three of them are collinear.

Along a round table are arranged $11$ cards with the names ( all distinct ) of the $11$ members of the $16^{th}$ JBMO Problem Selection Committee . The cards are arranged in a regular polygon manner . Assume that in the first meeting of the Committee none of its $11$ members sits in front of the card with his name . Is it possible to rotate the table by some angle so that at the end at least two members sit in front of the card with their names ?

What maximum number of elements can be selected from the set $\{1, 2, 3, \dots, 100\}$ so that [b]no[/b] sum of any three selected numbers is equal to a selected number? [i]Proposed by A. Golovanov[/i]

We colour every square of the $ 2009$ x $ 2009$ board with one of $ n$ colours (we do not have to use every colour). A colour is called connected if either there is only one square of that colour or any two squares of the colour can be reached from one another by a sequence of moves of a chess queen without intermediate stops at squares having another colour (a chess quen moves horizontally, vertically or diagonally). Find the maximum $ n$, such that for every colouring of the board at least on colour present at the board is connected.

Several dwarves were lined up in a row, and then they lined up in a row in a different order. Is it possible that exactly one third of the dwarves have both of their neighbours remained and exactly one third of the dwarves have only one of their neighbours remained, if the number of the dwarves is a) 6; b) 9?

On a line are given $n>3$ points. Find the number of colorings of these points in red and blue, such that in any set of consequent points the difference between the numbers of red and blue points does not exceed $2$.

One has marked $n$ points on a circle and has drawn a certain number of chords whose endpoints are the marked points. It turned out that the following property is satisfied: whenever any $2021$ drawn chords are removed one can join any two marked points by a broken line composed of some of the remaining drawn chords. Prove that one can remove some of the drawn chords so that at most $2022n$ chords remain and the property described above is preserved.

From a collection of $n$ persons $q$ distinct two-member teams are selected and ranked $1, \cdots, q$ (no ties). Let $m$ be the least integer larger than or equal to $2q/n$. Show that there are $m$ distinct teams that may be listed so that : [b](i)[/b] each pair of consecutive teams on the list have one member in common and [b](ii)[/b] the chain of teams on the list are in rank order. [i]Alternative formulation.[/i] Given a graph with $n$ vertices and $q$ edges numbered $1, \cdots , q$, show that there exists a chain of $m$ edges, $m \geq \frac{2q}{n}$ , each two consecutive edges having a common vertex, arranged monotonically with respect to the numbering.

In a group of mathematicians, every mathematician has some friends (the relation of friend is reciprocal). Prove that there exists a mathematician, such that the average of the number of friends of all his friends is no less than the average of the number of friends of all these mathematicians.