A paper tetrahedron is cut along some of so that it can be developed onto the plane. Could it happen that this development cannot be placed on the plane in one layer?

Maria and Bilyana play the following game. Maria has $2024$ fair coins and Bilyana has $2023$ fair coins. They toss every coin they have. Maria wins if she has strictly more heads than Bilyana, otherwise Bilyana wins. What is the probability of Maria winning this game?

Let $a_1,a_2,\cdots,a_n$ be integers such that $1=a_1\le a_2\le \cdots\le a_{2019}=99$. Find the minimum $f_0$ of the expression $$f=(a_1^2+a_2^2+\cdots+a_{2019}^2)-(a_1a_3+a_2a_4+\cdots+a_{2017}a_{2019}),$$ and determine the number of sequences $(a_1,a_2,\cdots,a_n)$ such that $f=f_0$.

$2021$ points are given on a circle. Each point is colored by one of the $1,2, \cdots ,k$ colors. For all points and colors $1\leq r \leq k$, there exist an arc such that at least half of the points on it are colored with $r$. Find the maximum possible value of $k$.

Let $ n$ and $ k$ be positive integers and let $ S$ be a set of $ n$ points in the plane such that [b]i.)[/b] no three points of $ S$ are collinear, and [b]ii.)[/b] for every point $ P$ of $ S$ there are at least $ k$ points of $ S$ equidistant from $ P.$ Prove that: \[ k < \frac {1}{2} \plus{} \sqrt {2 \cdot n} \]

Consider a finite set of points $T\in\mathbb{R}^n$ contained in the $n$-dimensional unit ball centered at the origin, and let $X$ be the convex hull of $T$. Prove that for all positive integers $k$ and all points $x\in X$, there exist points $t_1, t_2, \dots, t_k\in T$, not necessarily distinct, such that their centroid \[\frac{t_1+t_2+\dots+t_k}{k}\]has Euclidean distance at most $\frac{1}{\sqrt{k}}$ from $x$. (The $n$-dimensional unit ball centered at the origin is the set of points in $\mathbb{R}^n$ with Euclidean distance at most $1$ from the origin. The convex hull of a set of points $T\in\mathbb{R}^n$ is the smallest set of points $X$ containing $T$ such that each line segment between two points in $X$ lies completely inside $X$.)

Let $a_1,a_2,\cdots ,a_9$ be $9$ positive integers (not necessarily distinct) satisfying: for all $1\le i<j<k\le 9$, there exists $l (1\le l\le 9)$ distinct from $i,j$ and $j$ such that $a_i+a_j+a_k+a_l=100$. Find the number of $9$-tuples $(a_1,a_2,\cdots ,a_9)$ satisfying the above conditions.

In the county some pairs of towns connected by two-way non-stop flight. From any town we can flight to any other (may be not on one flight). Gives, that if we consider any cyclic (i.e. beginning and finish towns match) route, consisting odd number of flights, and close all flights of this route, then we can found two towns, such that we can't fly from one to other. Proved, that we can divided all country on $4$ regions, such that any flight connected towns from other regions.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.

A competition consists of $25$ sports, each awarding one gold medal to a winner. $25$ athletes participate, each in all $25$ sports. There are also $25$ experts, each of whom must predict the number of gold medals each athlete will win. In each prediction, the medal counts must be non-negative integers summing to $25$. An expert is called competent if they correctly guess the number of gold medals for at least one athlete. What is the maximum number \( k \) such that the experts can make their predictions so that at least \( k \) of them are guaranteed to be competent regardless of the outcome? \\

There are 8 points in the plane.we write down the area of each triangle having all vertices amoung these points(totally 56 numbers).Let them be $a_1,a_2,\dots a_{56}$.Prove that there is a choice of plus or minus such that: $$\pm a_1 \pm a_2 \dots \pm a_{56}=0$$

In each square of an $n\times{n}$ grid there is a lamp. If the lamp is touched it changes its state every lamp in the same row and every lamp in the same column (the one that are on are turned off and viceversa). At the begin, all the lamps are off. Show that lways is possible, with an appropriated sequence of touches, that all the the lamps on the board end on and find, in function of $n$ the minimal number of touches that are necessary to turn on every lamp.

Albatross and Frankinfueter are playing again: each of them takes turns choosing one point in the plane with integer coordinates and paint it in his favorite color. Albatross plays first. Someone wins as soon as there is a square with all four corners in the are colored in their own color. Does anyone has a winning strategy and if so, who?

Thirty-four countries participated in a jury session of the IMO, each represented by the leader and the deputy leader of the team. Before the meeting, some participants exchanged handshakes, but no team leader shook hands with his deputy. After the meeting, the leader of the Illyrian team asked every other participant the number of people they had shaken hands with, and all the answers she got were different. How many people did the deputy leader of the Illyrian team greet ?

The siblings Robb, Arya and Sansa have received seven sealed bags from an unknown donor with varying number of beads. Six of the bags have labels indicating the number beads: $7, 9, 11, 13, 15, 18$, but the seventh bag lacks etiquette. The sensor has set certain requirements: Robb must have three bags and his sisters two bags each. In addition, Arya will have the bag that contains $7$ beads. The bags should be distributed so that each of the siblings get the same number of pearls (this is possible according to the donor). How many pearls are there in the bag without a label, how many pearls are there in total and how should the bags be distributed?

Let $S$ be a finite set of points in the plane, such that for each $2$ points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon all of whose vertices are contained in $S$. Find all possible values for the number of elements of $S$. Proposed by [i]Viktor Simjanoski, Macedonia[/i]

Let $S$ be a finite set. For a positive integer $n$, we say that a function $f\colon S\to S$ is an [i]$n$-th power[/i] if there exists some function $g\colon S\to S$ such that \[ f(x) = \underbrace{g(g(\ldots g(x)\ldots))}_{\mbox{\scriptsize $g$ applied $n$ times}} \] for each $x\in S$. Suppose that a function $f\colon S\to S$ is an $n$-th power for each positive integer $n$. Is it necessarily true that $f(f(x)) = f(x)$ for each $x\in S$?

Let $n \geq 2$ and $m$ be positive integers. $m$ ballot boxes are placed in a line. Two players $A$ and $B$ play by turns, beginning with $A$, in the following manner. Each turn, $A$ chooses two boxes and places a ballot in each of them. Afterwards, $B$ chooses one of the boxes, and removes every ballot from it. $A$ wins if after some turn of $B$, there exists a box containing $n$ ballots. For each $n$, find the minimum value of $m$ such that $A$ can guarantee a win independently of how $B$ plays.

Let $n$ be a positive integer, and let $A$ be a subset of $\{ 1,\cdots ,n\}$. An $A$-partition of $n$ into $k$ parts is a representation of n as a sum $n = a_1 + \cdots + a_k$, where the parts $a_1 , \cdots , a_k $ belong to $A$ and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set $\{ a_1 , a_2 , \cdots , a_k \} $. We say that an $A$-partition of $n$ into $k$ parts is optimal if there is no $A$-partition of $n$ into $r$ parts with $r<k$. Prove that any optimal $A$-partition of $n$ contains at most $\sqrt[3]{6n}$ different parts.

Georg has a $4$-sided die with the numbers $2, 3, 4$ and $5$. He rolls the die $17$ times and records the result of each roll on a board, so that eventually $17$ numbers are written on it. Georg notices that the average of the $17$ numbers is an integer. Is it possible that each of the numbers $2, 3, 4$ and $5$ appears at least three times on Georg’s board?

Given is an $n\times n$ board, with an integer written in each grid. For each move, I can choose any grid, and add $1$ to all $2n-1$ numbers in its row and column. Find the largest $N(n)$, such that for any initial choice of integers, I can make a finite number of moves so that there are at least $N(n)$ even numbers on the board.

In the plane are $10$ lines in general position, which means that no $2$ are parallel and no $3$ are concurrent. Where $2$ lines intersect, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?

$66$ points are given on a plane; collinearity is allowed. There are [b]exactly[/b] $2021$ lines passing by at least two of the given points. Determine the greatest number of points in a same line. Give an example.

Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.