Found problems: 1800
2014 China National Olympiad, 3
For non-empty number sets $S, T$, define the sets $S+T=\{s+t\mid s\in S, t\in T\}$ and $2S=\{2s\mid s\in S\}$.
Let $n$ be a positive integer, and $A, B$ be two non-empty subsets of $\{1,2\ldots,n\}$. Show that there exists a subset $D$ of $A+B$ such that
1) $D+D\subseteq 2(A+B)$,
2) $|D|\geq\frac{|A|\cdot|B|}{2n}$,
where $|X|$ is the number of elements of the finite set $X$.
2010 Contests, 3
Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?
2004 IberoAmerican, 3
Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$.
Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$.
Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.
1971 IMO Longlists, 10
In how many different ways can three knights be placed on a chessboard so that the number of squares attacked would be maximal?
2006 Iran Team Selection Test, 6
Let $G$ be a tournoment such that it's edges are colored either red or blue.
Prove that there exists a vertex of $G$ like $v$ with the property that, for every other vertex $u$ there is a mono-color directed path from $v$ to $u$.
1977 IMO Longlists, 51
Several segments, which we shall call white, are given, and the sum of their lengths is $1$. Several other segments, which we shall call black, are given, and the sum of their lengths is $1$. Prove that every such system of segments can be distributed on the segment that is $1.51$ long in the following way: Segments of the same colour are disjoint, and segments of different colours are either disjoint or one is inside the other. Prove that there exists a system that cannot be distributed in that way on the segment that is $1.49$ long.
2010 Polish MO Finals, 1
The integer number $n > 1$ is given and a set $S \subset \{0, 1, 2, \ldots, n-1\}$ with $|S| > \frac{3}{4} n$. Prove that there exist integer numbers $a, b, c$ such that the remainders after the division by $n$ of the numbers:
\[a, b, c, a+b, b+c, c+a, a+b+c\]
belong to $S$.
2023 Bundeswettbewerb Mathematik, 4
Exactly $n$ chords (i.e. diagonals and edges) of a regular $2n$-gon are coloured red, satisfying the following two conditions:
(1) Each of the $2n$ vertices occurs exactly once as the endpoint of a red chord.
(2) No two red chords have the same length.
For which positive integers $n \ge 2$ is this possible?
2012 ELMO Shortlist, 2
Determine whether it's possible to cover a $K_{2012}$ with
a) 1000 $K_{1006}$'s;
b) 1000 $K_{1006,1006}$'s.
[i]David Yang.[/i]
2010 Romania Team Selection Test, 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]
2004 Tuymaada Olympiad, 4
There are many opposition societies in the city of N.
Each society consists of $10$ members. It is known that for every $2004$ societies there is a person belonging to at least $11$ of them.
Prove that the government can arrest $2003$ people so that at least one member of each society is arrested.
[i]Proposed by V.Dolnikov, D.Karpov[/i]
2011 All-Russian Olympiad, 4
There are some counters in some cells of $100\times 100$ board. Call a cell [i]nice[/i] if there are an even number of counters in adjacent cells. Can exactly one cell be [i]nice[/i]?
[i]K. Knop[/i]
2009 Junior Balkan MO, 4
Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.
2005 Bundeswettbewerb Mathematik, 4
For any integer $n\geq 3$, let $A\left(n\right)$ denote the maximal number of self-intersections a closed broken line $P_1P_2...P_nP_1$ can have; hereby, we assume that no three vertices of the broken line $P_1P_2...P_nP_1$ are collinear.
Prove that
[b](a)[/b] if n is odd, then $A\left(n\right)=\frac{n\left(n-3\right)}{2}$;
[b](b)[/b] if n is even, then $A\left(n\right)=\frac{n\left(n-4\right)}{2}+1$.
[i]Note.[/i] A [i]self-intersection[/i] of a broken line is a (non-ordered) pair of two distinct non-adjacent segments of the broken line which have a common point.
2015 China Team Selection Test, 1
For a positive integer $n$, and a non empty subset $A$ of $\{1,2,...,2n\}$, call $A$ good if the set $\{u\pm v|u,v\in A\}$ does not contain the set $\{1,2,...,n\}$. Find the smallest real number $c$, such that for any positive integer $n$, and any good subset $A$ of $\{1,2,...,2n\}$, $|A|\leq cn$.
2003 All-Russian Olympiad, 3
Is it possible to write a natural number in every cell of an infinite chessboard in such a manner that for all integers $m, n > 100$, the sum of numbers in every $m\times n$ rectangle is divisible by $m + n \ ?$
2006 Bulgaria National Olympiad, 1
Consider the set $A=\{1,2,3\ldots ,2^n\}, n\ge 2$. Find the number of subsets $B$ of $A$ such that for any two elements of $A$ whose sum is a power of $2$ exactly one of them is in $B$.
[i]Aleksandar Ivanov[/i]
2003 Turkey Team Selection Test, 6
For all positive integers $n$, let $p(n)$ be the number of non-decreasing sequences of positive integers such that for each sequence, the sum of all terms of the sequence is equal to $n$. Prove that
\[\dfrac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.\]
2011 Benelux, 4
Abby and Brian play the following game: They first choose a positive integer $N$. Then they write numbers on a blackboard in turn. Abby starts by writing a $1$. Thereafter, when one of them has written the number $n$, the other writes down either $n + 1$ or $2n$, provided that the number is not greater than $N$. The player who writes $N$ on the blackboard wins.
(a) Determine which player has a winning strategy if $N = 2011$.
(b) Find the number of positive integers $N\leqslant2011$ for which Brian has a winning strategy.
(This is based on ISL 2004, Problem C5.)
2008 Romanian Master of Mathematics, 4
Consider a square of sidelength $ n$ and $ (n\plus{}1)^2$ interior points. Prove that we can choose $ 3$ of these points so that they determine a triangle (eventually degenerated) of area at most $ \frac12$.
2010 Iran Team Selection Test, 7
Without lifting pen from paper, we draw a polygon in such away that from every two adjacent sides one of them is vertical.
In addition, while drawing the polygon all vertical sides have been drawn from up to down. Prove that this polygon has cut itself.
2004 Iran Team Selection Test, 5
This problem is generalization of [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=5918]this one[/url].
Suppose $G$ is a graph and $S\subset V(G)$. Suppose we have arbitrarily assign real numbers to each element of $S$. Prove that we can assign numbers to each vertex in $G\backslash S$ that for each $v\in G\backslash S$ number assigned to $v$ is average of its neighbors.
2000 Bulgaria National Olympiad, 3
Let $A$ be the set of all binary sequences of length $n$ and denote $o =(0, 0, \ldots , 0) \in A$. Define the addition on $A$ as $(a_1, \ldots , a_n)+(b_1, \ldots , b_n) =(c_1, \ldots , c_n)$, where $c_i = 0$ when $a_i = b_i$ and $c_i = 1$ otherwise. Suppose that $f\colon A \to A$ is a function such that $f(0) = 0$, and for each $a, b \in A$, the sequences $f(a)$ and $f(b)$ differ in exactly as many places as $a$ and $b$ do. Prove that if $a$ , $b$, $c \in A$ satisfy $a+ b + c = 0$, then $f(a)+ f(b) + f(c) = 0$.
2016 Indonesia TST, 4
We call a subset $B$ of natural numbers [i]loyal[/i] if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all [i]loyal[/i] sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set
\[f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.\] Furthermore, we define \[F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).\] Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$. (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero).
[i]Proposed by Javad Abedi[/i]
2010 Contests, 2
There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.