Found problems: 1800
2003 Costa Rica - Final Round, 5
Each of the squares of an $8 \times 8$ board can be colored white or black. Find the number of colorings of the board such that every $2 \times 2$ square contains exactly 2 black squares and 2 white squares.
2012 ELMO Shortlist, 1
Let $n\ge2$ be a positive integer. Given a sequence $\left(s_i\right)$ of $n$ distinct real numbers, define the "class" of the sequence to be the sequence $\left(a_1,a_2,\ldots,a_{n-1}\right)$, where $a_i$ is $1$ if $s_{i+1} > s_i$ and $-1$ otherwise.
Find the smallest integer $m$ such that there exists a sequence $\left(w_i\right)$ of length $m$ such that for every possible class of a sequence of length $n$, there is a subsequence of $\left(w_i\right)$ that has that class.
[i]David Yang.[/i]
2014 China Team Selection Test, 2
Let $A_1A_2...A_{101}$ be a regular $101$-gon, and colour every vertex red or blue. Let $N$ be the number of obtuse triangles satisfying the following: The three vertices of the triangle must be vertices of the $101$-gon, both the vertices with acute angles have the same colour, and the vertex with obtuse angle have different colour.
$(1)$ Find the largest possible value of $N$.
$(2)$ Find the number of ways to colour the vertices such that maximum $N$ is acheived. (Two colourings a different if for some $A_i$ the colours are different on the two colouring schemes).
2002 Tournament Of Towns, 2
[list]
[*] A test was conducted in class. It is known that at least $\frac{2}{3}$ of the problems were hard. Each such problems were not solved by at least $\frac{2}{3}$ of the students. It is also known that at least $\frac{2}{3}$ of the students passed the test. Each such student solved at least $\frac{2}{3}$ of the suggested problems. Is this possible?
[*] Previous problem with $\frac{2}{3}$ replaced by $\frac{3}{4}$.
[*] Previous problem with $\frac{2}{3}$ replaced by $\frac{7}{10}$.[/list]
2011 Canadian Mathematical Olympiad Qualification Repechage, 7
One thousand students participate in the $2011$ Canadian Closed Mathematics Challenge. Each student is assigned a unique three-digit identification number $abc,$ where each of $a, b$ and $c$ is a digit between $0$ and $9,$ inclusive. Later, when the contests are marked, a number of markers will be hired. Each of the markers will be given a unique two-digit identification number $xy,$ with each of $x$ and $y$ a digit between $0$ and $9,$ inclusive. Marker $xy$ will be able to mark any contest with an identification number of the form $xyA$ or $xAy$ or $Axy,$ for any digit $A.$ What is the minimum possible number of markers to be hired to ensure that all contests will be marked?
2011 Baltic Way, 7
Let $T$ denote the $15$-element set $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$. Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$.
2012 Iran Team Selection Test, 1
Is it possible to put $\binom{n}{2}$ consecutive natural numbers on the edges of a complete graph with $n$ vertices in a way that for every path (or cycle) of length $3$ where the numbers $a,b$ and $c$ are written on its edges (edge $b$ is between edges $c$ and $a$), $b$ is divisible by the greatest common divisor of the numbers $a$ and $c$?
[i]Proposed by Morteza Saghafian[/i]
1997 Austrian-Polish Competition, 3
Numbers $\frac{49}{1}, \frac{49}{2}, ... , \frac{49}{97}$ are writen on a blackboard. Each time, we can replace two numbers (like $a, b$) with $2ab-a-b+1$. After $96$ times doing that prenominate action, one number will be left on the board. Find all the possible values fot that number.
2021 German National Olympiad, 3
For a fixed $k$ with $4 \le k \le 9$ consider the set of all positive integers with $k$ decimal digits such that each of the digits from $1$ to $k$ occurs exactly once.
Show that it is possible to partition this set into two disjoint subsets such that the sum of the cubes of the numbers in the first set is equal to the sum of the cubes in the second set.
1977 IMO Longlists, 36
Consider a sequence of numbers $(a_1, a_2, \ldots , a_{2^n}).$ Define the operation
\[S\biggl((a_1, a_2, \ldots , a_{2^n})\biggr) = (a_1a_2, a_2a_3, \ldots , a_{2^{n-1}a_{2^n}, a_{2^n}a_1).}\]
Prove that whatever the sequence $(a_1, a_2, \ldots , a_{2^n})$ is, with $a_i \in \{-1, 1\}$ for $i = 1, 2, \ldots , 2^n,$ after finitely many applications of the operation we get the sequence $(1, 1, \ldots, 1).$
2009 Turkey MO (2nd round), 3
[i]Alice[/i], who works for the [i]Graph County Electric Works[/i], is commissioned to wire the newly erected utility poles in $k$ days. Each day she either chooses a pole and runs wires from it to as many poles as she wishes, or chooses at most $17$ pairs of poles and runs wires between each pair. [i]Bob[/i], who works for the [i]Graph County Paint Works[/i], claims that, no matter how many poles there are and how [i]Alice[/i] connects them, all the poles can be painted using not more than $2009$ colors in such a way that no pair of poles connected by a wire is the same color. Determine the greatest value of $k$ for which [i]Bob[/i]'s claim is valid.
2007 Middle European Mathematical Olympiad, 2
A set of balls contains $ n$ balls which are labeled with numbers $ 1,2,3,\ldots,n.$ We are given $ k > 1$ such sets. We want to colour the balls with two colours, black and white in such a way, that
(a) the balls labeled with the same number are of the same colour,
(b) any subset of $ k\plus{}1$ balls with (not necessarily different) labels $ a_{1},a_{2},\ldots,a_{k\plus{}1}$ satisfying the condition $ a_{1}\plus{}a_{2}\plus{}\ldots\plus{}a_{k}\equal{} a_{k\plus{}1}$, contains at least one ball of each colour.
Find, depending on $ k$ the greatest possible number $ n$ which admits such a colouring.
2014 Bulgaria National Olympiad, 3
A real number $f(X)\neq 0$ is assigned to each point $X$ in the space.
It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have :
\[ f(O)=f(A)f(B)f(C)f(D). \]
Prove that $f(X)=1$ for all points $X$.
[i]Proposed by Aleksandar Ivanov[/i]
2001 Hungary-Israel Binational, 2
Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$.
If $n \geq 5$ and $e(G_{n}) \geq \frac{n^{2}}{4}+2$, prove that $G_{n}$ contains two triangles that share exactly one vertex.
2014 China Team Selection Test, 2
Let $A_1A_2...A_{101}$ be a regular $101$-gon, and colour every vertex red or blue. Let $N$ be the number of obtuse triangles satisfying the following: The three vertices of the triangle must be vertices of the $101$-gon, both the vertices with acute angles have the same colour, and the vertex with obtuse angle have different colour.
$(1)$ Find the largest possible value of $N$.
$(2)$ Find the number of ways to colour the vertices such that maximum $N$ is acheived. (Two colourings a different if for some $A_i$ the colours are different on the two colouring schemes).
2007 Baltic Way, 8
Call a set $A$ of integers [i]non-isolated[/i], if for every $a\in A$ at least one of the numbers $a-1$ and $a+1$ also belongs to $A$. Prove that the number of five-element non-isolated subsets of $\{1, 2,\ldots ,n\}$ is $(n-4)^2$.
2004 Romania Team Selection Test, 5
A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors).
Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.)
[hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]
1988 Romania Team Selection Test, 3
Consider all regular convex and star polygons inscribed in a given circle and having $n$ [i]sides[/i]. We call two such polygons to be equivalent if it is possible to obtain one from the other using a rotation about the center of the circle. How many classes of such polygons exist?
[i]Mircea Becheanu[/i]
1997 Romania Team Selection Test, 3
Let $n\ge 4$ be a positive integer and let $M$ be a set of $n$ points in the plane, where no three points are collinear and not all of the $n$ points being concyclic. Find all real functions $f:M\to\mathbb{R}$ such that for any circle $\mathcal{C}$ containing at least three points from $M$, the following equality holds:
\[\sum_{P\in\mathcal{C}\cap M} f(P)=0\]
[i]Dorel Mihet[/i]
2010 Pan African, 1
Seven distinct points are marked on a circle of circumference $c$. Three of the points form an equilateral triangle and the other four form a square. Prove that at least one of the seven arcs into which the seven points divide the circle has length less than or equal $\frac{c}{24}$.
2011 ELMO Shortlist, 4
Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal.
[i]David Yang.[/i]
2010 Iran MO (3rd Round), 1
Prove that the group of orientation-preserving symmetries of the cube is isomorphic to $S_4$ (the group of permutations of $\{1,2,3,4\}$).(20 points)
2008 China Western Mathematical Olympiad, 3
For a given positive integer $n$, find the greatest positive integer $k$, such that there exist three sets of $k$ non-negative distinct integers, $A=\{x_1,x_2,\cdots,x_k\}, B=\{y_1,y_2,\cdots,y_k\}$ and $C=\{z_1,z_2,\cdots,z_k\}$ with $ x_j\plus{}y_j\plus{}z_j\equal{}n$ for any $ 1\leq j\leq k$.
[size=85][color=#0000FF][Moderator edit: LaTeXified][/color][/size]
2007 China Second Round Olympiad, 2
In a $7\times 8$ chessboard, $56$ stones are placed in the squares. Now we have to remove some of the stones such that after the operation, there are no five adjacent stones horizontally, vertically or diagonally. Find the minimal number of stones that have to be removed.
2005 All-Russian Olympiad Regional Round, 8.8
8.8, 9.8, 11.8
a) 99 boxes contain apples and oranges. Prove that we can choose 50 boxes in such a way that they contain at least half of all apples and half of all oranges.
b) 100 boxes contain apples and oranges. Prove that we can choose 34 boxes in such a way that they contain at least a third of all apples and a third of all oranges.
c) 100 boxes contain apples, oranges and bananas. Prove that we can choose 51 boxes in such a way that they contain at least half of all apples, and half of all oranges and half of all bananas.
([i]I. Bogdanov, G. Chelnokov, E. Kulikov[/i])