Prove that given $n$ points in the plane, not all aligned, there exists a line that passes through exactly two of them. [hide=original wording]Demostrar que dados n puntos en el plano, no todos alineados, existe una recta que pasa por exactamente dos de ellos.[/hide]

Let $A$ and $B$ be two sets of non-negative integers, define $A+B$ as the set of the values obtained when we sum any (one) element of the set $A$ with any (one) element of the set $B$. For instance, if $A=\{2,3\}$ and $B=\{0,1,2,5\}$ so $A+B=\{2,3,4,5,7,8\}$. Determine the least integer $k$ such that there is a pair of sets $A$ and $B$ of non-negative integers with $k$ and $2k$ elements, respectively, and $A+B=\{0,1,2,\dots, 2019,2020\}$

Consider the following operation on positive real numbers written on a blackboard: Choose a number $ r$ written on the blackboard, erase that number, and then write a pair of positive real numbers $ a$ and $ b$ satisfying the condition $ 2 r^2 \equal{} ab$ on the board. Assume that you start out with just one positive real number $ r$ on the blackboard, and apply this operation $ k^2 \minus{} 1$ times to end up with $ k^2$ positive real numbers, not necessarily distinct. Show that there exists a number on the board which does not exceed kr.

At the beginning of school year in one of the first grade classes: $i)$ every student had exatly $20$ acquaintances $ii)$ every two students knowing each other had exactly $13$ mutual acquaintances $iii)$ every two students not knowing each other had exactly $12$ mutual acquaintances Find number of students in this class

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

[b]E[/b]ach of the integers $1,2,...,729$ is written in its base-$3$ representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122...$ How many times in this string does the substring $012$ appear?

On the numerical line, arrange a system of closed segments of length $1$ without common points (endpoints included) so that any infinite arithmetic progression with any non zero difference and any first term has a common point with a segment of the system.

Some points in the plane are either colored red or blue. The distance between two points of the opposite color is at most 1. Prove that there exists a circle with diameter $\sqrt{2}$ such that no two points outside of this circle have same color. It is enough to prove this claim for a finite number of colored points.

Let $M$ be the set of all integers from $1$ to $2013$. Each subset of $M$ is given one of $k$ available colors, with the only condition that if the union of two different subsets $A$ and $B$ is $M$, then $A$ and $B$ are given different colors. What is the least possible value of $k$?

Suppose that $ p_1,p_2,p_3,q_1,q_2,q_3$ are six points in the plane and that the distance between $ p_i$ and $ q_j$ ($ i,j \equal{} 1,2,3$) is $ i \plus{} j$. Show that the six points are collinear.

The points in the plane are painted in 100 colors. Prove that there are three points of the same color that are the vertices of a triangle of area 1. [i]Proposed by V. Bragin[/i]

Every integer $> 2024$ is given a color, white or black. The product of any two white integers is a black integer. Prove that there are two black integers that have a difference of one.

We are given a bag of sugar, a two-pan balance, and a weight of $1$ gram. How do we obtain $1$ kilogram of sugar in the smallest possible number of weighings?

Some pairs of cities in the country are connected by airlines, and some are not. But every city has an airport, from which you can get to any other city, making no more than one transfer. A tourist who wants to make a round trip through several cities of the country will have to fly around at least five cities. Prove that the same number of airlines depart from each city of the country (If there is an airline from one city to another, then there is also one from the second to the first. A circular trip is a route that passes through at least three cities, starting and ending in same city, other cities are not repeated in it)

Ayman wants to color the cells of a $50 \times 50$ chessboard into black and white so that each $2 \times 3$ or $3 \times 2$ rectangle contains an even number of white cells. Determine the number of ways Ayman can color the chessboard.

Call an $n$-gon to be [i]lattice[/i] if its vertices are lattice points. Prove that inside every lattice convex pentagon there exists a lattice point.

A regular hexagon with side length $ 1 $ is given. There are $ m $ points in its interior such that no $ 3 $ are collinear. The hexagon is divided into triangles (triangulated), such that every point of the $ m $ given and every vertex of the hexagon is a vertex of such a triangle. The triangles don't have common interior points. Prove that there exists a triangle with area not greater than $ \frac{3 \sqrt{3}}{4(m+2)}$.

In a plane, there is an infinite triangular grid consists of equilateral triangles whose lengths of the sides are equal to $ 1$, call the vertices of the triangles the lattice points, call two lattice points are adjacent if the distance between the two points is equal to $ 1;$ A jump game is played by two frogs $ A,B,$ "A jump" is called if the frogs jump from the point which it is lying on to its adjacent point, " A round jump of $ A,B$" is called if first $ A$ jumps and then $ B$ by the following rules: Rule (1): $ A$ jumps once arbitrarily, then $ B$ jumps once in the same direction, or twice in the opposite direction; Rule (2): when $ A,B$ sits on adjacent lattice points, they carry out Rule (1) finishing a round jump, or $ A$ jumps twice continually, keep adjacent with $ B$ every time, and $ B$ rests on previous position; If the original positions of $ A,B$ are adjacent lattice points, determine whether for $ A$ and $ B$,such that the one can exactly land on the original position of the other after a finite round jumps.

There are dwarves in a forest and each one of them owns exactly 3 hats which are numbered with numbers $1, 2, \dots 28$. Three hats of a dwarf are numbered with different numbers and there are 3 festivals in this forest in a day. In the first festival, each dwarf wears the hat which has the smallest value, in the second festival, each dwarf wears the hat which has the second smallest value and in the final festival each dwarf wears the hat which has the biggest value. After that, it is realized that there is no dwarf pair such that both of two dwarves wear the same value in at least two festivals. Find the maximum value of number of dwarves.

In the Euclidean three-dimensional space, we call [i]folding[/i] of a sphere $S$ every partition of $S \setminus \{x,y\}$ into disjoint circles, where $x$ and $y$ are two points of $S$. A folding of $S$ is called [b]linear[/b] if the circles of the [i]folding[/i] are obtained by the intersection of $S$ with a family of parallel planes or with a family of planes containing a straight line $D$ exterior to $S$. Is every [i]folding[/i] of a sphere $S$ [b]linear[/b]?

Let $n$ be a posititve integer and let $M$ the set of all all integer cordinates $(a,b,c)$ such that $0 \le a,b,c \le n$. A frog needs to go from the point $(0,0,0)$ to the point $(n,n,n)$ with the following rules: $\cdot$ The frog can jump only in points of $M$ $\cdot$ The frog can't jump more than $1$ time over the same point. $\cdot$ In each jump the frog can go from $(x,y,z)$ to $(x+1,y,z)$, $(x,y+1,z)$, $(x,y,z+1)$ or $(x,y,z-1)$ In how many ways the Frog can make his target?

Given an expression $$x_1 : x_2 : ... : x_n$$ ( $:$ means division). We can put the braces as we want. How many expressions can we obtain?

Consider a board with $n$ rows and $4$ columns. In the first line are written $4$ zeros (one in each house). Next, each line is then obtained from the previous line by performing the following operation: one of the houses, (that you can choose), is maintained as in the previous line; the other three are changed: * if in the previous line there was a $0$, then in the down square $1$ is placed; * if in the previous line there was a $1$, then in the down square $2$ is placed; * if in the previous line there was a $2$, then in the down square $0$ is placed; Build the largest possible board with all its distinct lines and demonstrate that it is impossible to build a larger board.

There are $11$ empty boxes. In one move you can put one coin in some 10 of them. Two people play and take turns. Wins the one after which for the first time there will be $21$ coins in one of the boxes. Who wins when played correctly?

Find the number of $(a_1,a_2, ... ,a_{2014})$ permutations of the $(1,2, . . . ,2014)$ such that, for all $1\leq i<j\leq2014$, $i+a_i \leq j+a_j$.