Ten closed intervals, each of length $1$, are placed in the interval $[0,4]$. Show that there is a point in the larger interval that belongs to at least four of the smaller intervals.

An $n\times n$ table whose cells are numerated with numbers $1, 2,\cdots, n^2$ in some order is called [i]Naissus[/i] if all products of $n$ numbers written in $n$ [i]scattered[/i] cells give the same residue when divided by $n^2+1$. Does there exist a Naissus table for $(a) n = 8;$ $(b) n = 10?$ ($n$ cells are [i]scattered[/i] if no two are in the same row or column.) [i]Proposed by Marko Djikic[/i]

A number of robots are placed on the squares of a finite, rectangular grid of squares. A square can hold any number of robots. Every edge of each square of the grid is classified as either passable or impassable. All edges on the boundary of the grid are impassable. You can give any of the commands up, down, left, or right. All of the robots then simultaneously try to move in the specified direction. If the edge adjacent to a robot in that direction is passable, the robot moves across the edge and into the next square. Otherwise, the robot remains on its current square. You can then give another command of up, down, left, or right, then another, for as long as you want. Suppose that for any individual robot, and any square on the grid, there is a finite sequence of commands that will move that robot to that square. Prove that you can also give a finite sequence of commands such that all of the robots end up on the same square at the same time.

Consider the set $S = {1,2,3,\ldots , j}$. Let $m(A)$ denote the maximum element of $A$. Prove that $$\sum_ {A\subseteq S} m(A) = (j-1)2^j +1$$

A square-shaped pizza with side length $30$ cm is cut into pieces (not necessarily rectangular). All cuts are parallel to the sides, and the total length of the cuts is $240$ cm. Show that there is a piece whose area is at least $36$ cm$^2$

We are given more than $2^k$ integers, where $k\in\mathbb{N}$. Prove that we can choose $k+2$ of them such that if some of our selected numbers satisfy \[x_1+x_2+\dots+x_m=y_1+y_2+\dots+y_m\] where $x_1<\dots<x_m$ and $y_1<\dots<y_m$, then $x_i=y_i$ for any $1\le i\le m$.

A stromino is a $3 \times 1$ rectangle. Show that a $5 \times 5$ board divided into twenty-five $1 \times 1$ squares cannot be covered by $16$ strominos such that each stromino covers exactly three squares of the board, and every square is covered by one or two strominos. (A stromino can be placed either horizontally or vertically on the board.) [i]Proposed by Navilarekallu Tejaswi[/i]

[b]p1.[/b] In the group of five people any subgroup of three persons contains at least two friends. Is it possible to divide these five people into two subgroups such that all members of any subgroup are friends? [b]p2.[/b] Coefficients $a,b,c$ in expression $ax^2+bx+c$ are such that $b-c>a$ and $a \ne 0$. Is it true that equation $ax^2+bx+c=0$ always has two distinct real roots? [b]p3.[/b] Point $D$ is a midpoint of the median $AF$ of triangle $ABC$. Line $CD$ intersects $AB$ at point $E$. Distances $|BD|=|BF|$. Show that $|AE|=|DE|$. [b]p4.[/b] Real numbers $a,b$ satisfy inequality $a+b^5>ab^5+1$. Show that $a+b^7>ba^7+1$. [b]p5.[/b] A positive number was rounded up to the integer and got the number that is bigger than the original one by $28\%$. Find the original number (find all solutions). [b]p6.[/b] Divide a $5\times 5$ square along the sides of the cells into $8$ parts in such a way that all parts are different. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We color one of the numbers $1,...,8$ with white or black according to the following rules: i) number $4$ gets colored white and one at lest of the following numbers gets colored black ii) if two numbers $a,b$ are colored in a different color and $a+b\le 8$, then number $a+b$ gets colored black. iii) if two numbers $a,b$ are colored in a different color and $a\cdot b\le 8$, then number $a\cdot b$ gets colored white. If by those rules, all numbers get colored, find the color of each number.

In a sequence $a_1, a_2, . . . , a_{1000}$ consisting of $1000$ distinct numbers a pair $(a_i, a_j )$ with $i < j$ is called [i]ascending [/i] if $a_i < a_j$ and [i]descending[/i] if $a_i > a_j$ . Determine the largest positive integer $k$ with the property that every sequence of $1000$ distinct numbers has at least $k$ non-overlapping ascending pairs or at least $k$ non-overlapping descending pairs.

Let $ n $ be a positive integer. A sequence $(a_1,a_2,\ldots,a_n)$ of length is called $balanced$ if for every $ k $ $(1\leq k\leq n)$ the term $ a_k $ is equal with the number of distinct numbers from the subsequence $(a_1,a_2,\ldots,a_k).$ a) How many balanced sequences $(a_1,a_2,\ldots,a_n)$ of length $ n $ do exist? b) For every positive integer $m$ find how many balanced sequences $(a_1,a_2,\ldots,a_n)$ of length $ n $ exist such that $a_n=m.$

Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$

Is it possible to cover the plane with (infinite) circles so that exactly $2020$ circles pass through each point on the plane?

Find the smallest positive integer $n$ with the following property: After painting black exactly $n$ cells of a $7\times 7$ board there always exists a $2\times 2$ square with at least three black cells.

In a very long corridor there is an infinite number of cabinets, which start with $1,2,3,...$ numbered and initially all are closed. There is also a horde of QEDlers, whose number lies in set $A \subseteq \{1, 2,3,...\}$ . In ascending order, the QED people now cause chaos: the person with number $a \in A$ visits the cabinet with the numbers $a,2a,3a,...$ opening all of the closed ones and closes all open. Show that in the end the cabinet has never exactly the same numbers from $A$ open.

There are $2005$ players in a chess tournament played a game. Every pair of players played a game against each other. At the end of the tournament, it turned out that if two players $A$ and $B$ drew the game between them, then every other player either lost to $A$ or to $B$. Suppose that there are at least two draws in the tournament. Prove that all players can be lined up in a single file from left to right in the such a way that every play won the game against the person immediately to his right.

There are $n$ cities in a country, where $n>1$. There are railroads connecting some of the cities so that you can travel between any two cities through a series of railroads (railroads run in both direction.) In addition, in this country, it is impossible to travel from a city, through a series of distinct cities, and return back to the original city. We define the [b]degree[/b] of a city as the number of cities directly connected to it by a single segment of railroad. For a city $A$ that is directly connected to $x$ cities, with $y$ of those cities having a smaller degree than city $A$, the [b]significance[/b] of city $A$ is defined as $\frac{y}{x}$. Find the smallest positive real number $t$ so that, for any $n>1$, the sum of the significance of all cities is less than $tn$, no matter how the railroads are paved. [i]Proposed by houkai[/i]

Legend has it that in a police station in the old west a group of six bandits tried to bribe the Sheriff in charge of the place with six gold coins to free them, the Sheriff was a very honest person so to prevent them from continuing to insist With the idea of bribery, he sat the $6$ bandits around a table and proposed the following: - "Initially the leader will have the six gold coins, in each turn one of you can pass coins to the adjacent companions, but each time you do so you must pass the same amount of coins to each of your neighbors. If at any time they all manage to have the same amount of coins so I will let them go free. " The bandits accepted and began to play. Show that regardless of what moves the bandits make, they cannot win.

Determine the number of sets $A = \{a_1,a_2,...,a_{1000}\}$ of positive integers satisfying $a_1 < a_2 <...< a_{1000} \le 2014$, for which we have that the set $S = \{a_i + a_j | 1 \le i, j \le 1000$ with $i + j \in A\}$ is a subset of $A$.

The vertices of the cube are assigned $1, 2, 3..., 8$ and then each edge we assign the product of the numbers assigned to its two extreme points. Determine the greatest possible the value of the sum of the numbers assigned to all twelve edges of the cube.

Let $n>k \geq 1$ be integers and let $p$ be a prime dividing $\tbinom{n}{k}$. Prove that the $k$-element subsets of $\{1,\ldots,n\}$ can be split into $p$ classes of equal size, such that any two subsets with the same sum of elements belong to the same class. [i]Ankan Bhattacharya[/i]

In an infinite grid of regular triangles, Niels and Henrik are playing a game they made up. Every other time, Niels picks a triangle and writes $\times$ in it, and every other time, Henrik picks a triangle where he writes a $o$. If one of the players gets four in a row in some direction (see figure), he wins the game. Determine whether one of the players can force a victory. [img]https://cdn.artofproblemsolving.com/attachments/6/e/5e80f60f110a81a74268fded7fd75a71e07d3a.png[/img]

Each set of a finite family of subsets of a line is a union of two closed intervals. Moreover, any three of the sets of the family have a point in common. Prove that there is a point which is common to at least half the sets of the family.

Given is a set of $n\ge5$ people and $m$ commissions with $3$ persons in each. Let all the commissions be [i]nice[/i] if there are no two commissions $A$ and $B$, such that $\mid A\cap B\mid=1$. Find the biggest possible $m$ (as a function of $n$).

Given right $n$-gon wit the point $O$ -- its centre. All the vertices are marked either with $+1$ or $-1$. We may change all the signs in the vertices of regular $k$-gon ($2 \le k \le n$) with the same centre $O$. (By $2$-gon we understand a segment, being halved by $O$.) Prove that in a), b) and c) cases there exists such a set of $(+1)$s and $(-1)$s, that we can never obtain a set of $(+1)$s only. a) $n = 15$, b) $n = 30$, c) $n > 2$, d) Let us denote $K(n)$ the maximal number of $(+1)$ and $(-1)$ sets such, that it is impossible to obtain one set from another. Prove, for example, that $K(200) = 2^{80}$