Found problems: 1800
2010 Contests, 1
Given an integer number $n \geq 3$, consider $n$ distinct points on a circle, labelled $1$ through $n$.
Determine the maximum number of closed chords $[ij]$, $i \neq j$, having pairwise non-empty intersections.
[i]János Pach[/i]
1970 IMO Longlists, 38
Find the greatest integer $A$ for which in any permutation of the numbers $1, 2, \ldots , 100$ there exist ten consecutive numbers whose sum is at least $A$.
2014 Contests, 3
There are $n$ students sitting on a round table. You collect all of $ n $ name tags and give them back arbitrarily. Each student gets one of $n$ name tags. Now $n$ students repeat following operation:
The students who have their own name tags exit the table. The other students give their name tags to the student who is sitting right to him.
Find the number of ways giving name tags such that there exist a student who don't exit the table after 4 operations.
2013 Balkan MO Shortlist, C1
In a mathematical competition, some competitors are friends; friendship is mutual, that is, when $A$ is a friend of $B$, then $B$ is also a friend of $A$.
We say that $n \geq 3$ different competitors $A_1, A_2, \ldots, A_n$ form a [i]weakly-friendly cycle [/i]if $A_i$ is not a friend of $A_{i+1}$ for $1 \leq i \leq n$ (where $A_{n+1} = A_1$), and there are no other pairs of non-friends among the components of the cycle.
The following property is satisfied:
"for every competitor $C$ and every weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element"
Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends.
([i]Serbia[/i])
2006 Pre-Preparation Course Examination, 6
Show that the product of every $k$ consecutive members of the Fibonacci sequence is divisible by $f_1f_2\ldots f_k$ (where $f_0=0$ and $f_1=1$).
2010 International Zhautykov Olympiad, 3
A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even.
[img]http://up.iranblog.com/Files7/dda310bab8b6455f90ce.jpg[/img]
2007 Polish MO Finals, 3
3. Plane is divided with horizontal and vertical lines into unit squares. Into each square we write a positive integer so that each positive integer appears exactly once. Determine whether it is possible to write numbers in such a way, that each written number is a divisor of a sum of its four neighbours.
2011 IMAC Arhimede, 3
Place $n$ points on a circle and draw all possible chord joining these points. If no three chord are concurent, find the number of disjoint regions created.
[color=#008000]Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=260926&hilit=circle+points+segments+regions[/color]
2008 Portugal MO, 6
Let $n$ be a natural number larger than $2$. Vanessa has $n$ piles of jade stones, and all the piles have a different number of stones. Vanessa can distribute the stones from any pile by the other piles and stay with $n-1$ piles with the same number of stones. She also can distribute the stones from any two piles by the other piles and stay with $n-2$ piles with the same number of stones. Find the smallest possible number of jade's stones that the pile with the largest number of stones can have.
2007 China Western Mathematical Olympiad, 1
Let set $ T \equal{} \{1,2,3,4,5,6,7,8\}$. Find the number of all nonempty subsets $ A$ of $ T$ such that $ 3|S(A)$ and $ 5\nmid S(A)$, where $ S(A)$ is the sum of all the elements in $ A$.
2004 Regional Olympiad - Republic of Srpska, 3
An $8\times8$ chessboard is completely tiled by $2\times1$ dominoes. Prove that we can place positive integers
in all cells of the table in such a way that the sums of numbers in every domino are equal and the numbers placed
in two adjacent cells are coprime if and only if they belong to the same domino. (Two cells are called adjacent if
they have a common side.)
Well this can belong to number theory as well...
2007 JBMO Shortlist, 2
Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.
2005 Mediterranean Mathematics Olympiad, 1
The professor tells Peter the product of two positive integers and Sam their sum. At first, nobody of them knows the number of the other.
One of them says: [i]You can't possibly guess my number[/i].
Then the other says: [i]You are wrong, the number is 136[/i].
Which number did the professor tell them respectively? Give reasons for your claim.
1995 Baltic Way, 11
In how many ways can the set of integers $\{1,2,\ldots ,1995\}$ be partitioned into three non-empty sets so that none of these sets contains any pair of consecutive integers?
2007 Bulgaria Team Selection Test, 4
Let $G$ is a graph and $x$ is a vertex of $G$. Define the transformation $\varphi_{x}$ over $G$ as deleting all incident edges with respect of $x$ and drawing the edges $xy$ such that $y\in G$ and $y$ is not connected with $x$ with edge in the beginning of the transformation. A graph $H$ is called $G-$[i]attainable[/i] if there exists a sequece of such transformations which transforms $G$ in $H.$ Let $n\in\mathbb{N}$ and $4|n.$ Prove that for each graph $G$ with $4n$ vertices and $n$ edges there exists $G-$[i]attainable[/i] graph with at least $9n^{2}/4$ triangles.
2020 Baltic Way, 9
Each vertex $v$ and each edge $e$ of a graph $G$ are assigned numbers $f(v)\in\{1,2\}$ and $f(e)\in\{1,2,3\}$, respectively.
Let $S(v)$ be the sum of numbers assigned to the edges incident to $v$ plus the number $f(v)$.
We say that an assignment $f$ is [i]cool [/i]if $S(u) \ne S(v)$ for every pair $(u,v)$ of adjacent (i.e. connected by an edge) vertices in $G$.
Prove that for every graph there exists a cool assignment.
2012 Bulgaria National Olympiad, 2
Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled:
1) For every prime number $p$ and every natural number $n$, the numbers $p^n,p^{n+1}$ and $p^{n+2}$ do not have the same colour.
2) There does not exist an infinite geometric sequence of natural numbers of the same colour.
2012 China Western Mathematical Olympiad, 3
Let $A$ be a set of $n$ elements and $A_1, A_2, ... A_k$ subsets of $A$ such that for any $2$ distinct subsets $A_i, A_j$ either they are disjoint or one contains the other. Find the maximum value of $k$
2010 Contests, 4
On the plane are given $ k\plus{}n$ distinct lines , where $ k>1$ is integer and $ n$ is integer as well.Any three of these lines do not pass through the
same point . Among these lines exactly $ k$ are parallel and all the other $ n$ lines intersect each other.All $ k\plus{}n$ lines define on the plane a partition
of triangular , polygonic or not bounded regions. Two regions are colled different, if the have not common points
or if they have common points only on their boundary.A regions is called ''good'' if it contained in a zone between two parallel lines .
If in a such given configuration the minimum number of ''good'' regionrs is $ 176$ and the maximum number of these regions is $ 221$, find $ k$ and $ n$.
Babis
2023 Mexico National Olympiad, 4
Let $n \ge 2$ be a positive integer. For every number from $1$ to $n$, there is a card with this number and which is either black or white.
A magician can repeatedly perform the following move: For any two tiles with different number and different colour, he can replace the card with the smaller number by one identical to the other card.
For instance, when $n=5$ and the initial configuration is $(1B, 2B, 3W, 4B,5B)$, the magician can choose $1B, 3W$ on the first move to obtain $(3W, 2B, 3W, 4B, 5B)$ and then $3W, 4B$ on the second move to obtain $(4B, 2B, 3W, 4B, 5B)$.
Determine in terms of $n$ all possible lengths of sequences of moves from any possible initial configuration to any configuration in which no more move is possible.
2009 Italy TST, 1
Let $n,k$ be positive integers such that $n\ge k$. $n$ lamps are placed on a circle, which are all off. In any step we can change the state of $k$ consecutive lamps. In the following three cases, how many states of lamps are there in all $2^n$ possible states that can be obtained from the initial state by a certain series of operations?
i)$k$ is a prime number greater than $2$;
ii) $k$ is odd;
iii) $k$ is even.
2013 Tuymaada Olympiad, 5
Each face of a $7 \times 7 \times 7$ cube is divided into unit squares. What is the maximum number of squares that can be chosen so that no two chosen squares have a common point?
[i]A. Chukhnov[/i]
2010 ELMO Shortlist, 8
A tree $T$ is given. Starting with the complete graph on $n$ vertices, subgraphs isomorphic to $T$ are erased at random until no such subgraph remains. For what trees does there exist a positive constant $c$ such that the expected number of edges remaining is at least $cn^2$ for all positive integers $n$?
[i]David Yang.[/i]
2005 Bundeswettbewerb Mathematik, 4
Prove that each finite set of integers can be arranged without intersection.
2012 Tuymaada Olympiad, 4
Integers not divisible by $2012$ are arranged on the arcs of an oriented graph. We call the [i]weight of a vertex [/i]the difference between the sum of the numbers on the arcs coming into it and the sum of the numbers on the arcs going away from it. It is known that the weight of each vertex is divisible by $2012$. Prove that non-zero integers with absolute values not exceeding $2012$ can be arranged on the arcs of this graph, so that the weight of each vertex is zero.
[i]Proposed by W. Tutte[/i]