Found problems: 1800
2003 Serbia Team Selection Test, 3
Each edge and each diagonal of the convex $ n$-gon $ (n\geq 3)$ is colored in red or blue. Prove that the vertices of the $ n$-gon can be labeled as $ A_1,A_2,...,A_n$ in such a way that one of the following two conditions is satisfied:
(a) all segments $ A_1A_2,A_2A_3,...,A_{n\minus{}1}A_n,A_nA_1$ are of the same colour.
(b) there exists a number $ k, 1<k<n$ such that the segments $ A_1A_2,A_2A_3,...,A_{k\minus{}1}A_k$ are blue, and the segments $ A_kA_{k\plus{}1},...,A_{n\minus{}1}A_n,A_nA_1$ are red.
2014 Miklós Schweitzer, 3
We have $4n + 5$ points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form $n$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occurring as vertices of the $n$ triangles have the same color.
2014 China Team Selection Test, 6
Let $n\ge 2$ be a positive integer. Fill up a $n\times n$ table with the numbers $1,2,...,n^2$ exactly once each. Two cells are termed adjacent if they have a common edge. It is known that for any two adjacent cells, the numbers they contain differ by at most $n$. Show that there exist a $2\times 2$ square of adjacent cells such that the diagonally opposite pairs sum to the same number.
2011 Iran MO (3rd Round), 3
Suppose that $p(n)$ is the number of partitions of a natural number $n$. Prove that there exists $c>0$ such that $P(n)\ge n^{c \cdot \log n}$.
[i]proposed by Mohammad Mansouri[/i]
2013 China Team Selection Test, 3
Let $A$ be a set consisting of 6 points in the plane. denoted $n(A)$ as the number of the unit circles which meet at least three points of $A$. Find the maximum of $n(A)$
2013 Romania Team Selection Test, 4
Let $k$ be a positive integer larger than $1$. Build an infinite set $\mathcal{A}$ of subsets of $\mathbb{N}$ having the following properties:
[b](a)[/b] any $k$ distinct sets of $\mathcal{A}$ have exactly one common element;
[b](b)[/b] any $k+1$ distinct sets of $\mathcal{A}$ have void intersection.
2014 Turkey MO (2nd round), 1
In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$. In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after $2014$ steps?
2014 National Olympiad First Round, 20
How many distinct sets are there such that each set contains only non-negative powers of $2$ or $3$ and sum of its elements is $2014$?
$
\textbf{(A)}\ 64
\qquad\textbf{(B)}\ 60
\qquad\textbf{(C)}\ 54
\qquad\textbf{(D)}\ 48
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2007 Turkey MO (2nd round), 3
In a country between each pair of cities there is at most one direct road. There is a connection (using one or more roads) between any two cities even after the elimination of any given city and all roads incident to this city. We say that the city $A$ can be[i] k -directionally[/i] connected to the city $B$, if : we can orient at most $k$ roads such that after[i] arbitrary[/i] orientation of remaining roads for any fixed road $l$ (directly connecting two cities) there is a path passing through roads in the direction of their orientation starting at $A$, passing through $l$ and ending at $B$ and visiting each city at most once. Suppose that in a country with $n$ cities, any two cities can be[i] k - directionally[/i] connected. What is the minimal value of $k$?
2009 Indonesia TST, 1
Given an $ n\times n$ chessboard.
a) Find the number of rectangles on the chessboard.
b) Assume there exists an $ r\times r$ square (label $ B$) with $ r<n$ which is located on the upper left corner of the board. Define "inner border" of $ A$ as the border of $ A$ which is not the border of the chessboard. How many rectangles in $ B$ that touch exactly one inner border of $ B$?
1983 IMO Longlists, 44
We are given twelve coins, one of which is a fake with a different mass from the other eleven. Determine that coin with three weighings and whether it is heavier or lighter than the others.
2014 Singapore MO Open, 4
Fill up each square of a $50$ by $50$ grid with an integer. Let $G$ be the configuration of $8$ squares obtained by taking a $3$ by $3$ grid and removing the central square. Given that for any such $G$ in the $50$ by $50$ grid, the sum of integers in its squares is positive, show there exist a $2$ by $2$ square such that the sum of its entries is also positive.
2006 Germany Team Selection Test, 2
In a room, there are $2005$ boxes, each of them containing one or several sorts of fruits, and of course an integer amount of each fruit.
[b]a)[/b] Show that we can find $669$ boxes, which altogether contain at least a third of all apples and at least a third of all bananas.
[b]b)[/b] Can we always find $669$ boxes, which altogether contain at least a third of all apples, at least a third of all bananas and at least a third of all pears?
2001 Baltic Way, 2
Let $n\ge 2$ be a positive integer. Find whether there exist $n$ pairwise nonintersecting nonempty subsets of $\{1, 2, 3, \ldots \}$ such that each positive integer can be expressed in a unique way as a sum of at most $n$ integers, all from different subsets.
1985 IMO Longlists, 48
In a given country, all inhabitants are knights or knaves. A knight never lies; a knave always lies. We meet three persons, $A, B$, and $C$. Person $A$ says, “If $C$ is a knight, $B$ is a knave.” Person $C$ says, “$A$ and I are different; one is a knight and the other is a knave.” Who are the knights, and who are the knaves ?
2010 ELMO Shortlist, 6
Hamster is playing a game on an $m \times n$ chessboard. He places a rook anywhere on the board and then moves it around with the restriction that every vertical move must be followed by a horizontal move and every horizontal move must be followed by a vertical move. For what values of $m,n$ is it possible for the rook to visit every square of the chessboard exactly once? A square is only considered visited if the rook was initially placed there or if it ended one of its moves on it.
[i]Brian Hamrick.[/i]
1972 Bundeswettbewerb Mathematik, 4
$p>2$ persons participate at a chess tournament, two players play at most one game against each other. After $n$ games were made, no more game is running and in every subset of three players, we can find at least two that havem't played against each other.
Show that $n \leq \frac{p^{2}}4$.
2007 Junior Balkan Team Selection Tests - Romania, 4
We call a set of points [i]free[/i] if there is no equilateral triangle with the vertices among the points of the set. Prove that every set of $n$ points in the plane contains a [i]free[/i] subset with at least $\sqrt{n}$ elements.
2011 Korea National Olympiad, 3
There are $n$ students each having $r$ positive integers. Their $nr$ positive integers are all different. Prove that we can divide the students into $k$ classes satisfying the following conditions.
(a) $ k \le 4r $
(b) If a student $A$ has the number $m$, then the student $B$ in the same class can't have a number $l$ such that
\[ (m-1)! < l < (m+1)!+1 \]
2013 Greece National Olympiad, 3
We define the sets $A_1,A_2,...,A_{160}$ such that $\left|A_{i} \right|=i$ for all $i=1,2,...,160$. With the elements of these sets we create new sets $M_1,M_2,...M_n$ by the following procedure: in the first step we choose some of the sets $A_1,A_2,...,A_{160}$ and we remove from each of them the same number of elements. These elements that we removed are the elements of $M_1$. In the second step we repeat the same procedure in the sets that came of the implementation of the first step and so we define $M_2$. We continue similarly until there are no more elements in $A_1,A_2,...,A_{160}$, thus defining the sets $M_1,M_2,...,M_n$. Find the minimum value of $n$.
2012 Romania Team Selection Test, 4
Let $S$ be a set of positive integers, each of them having exactly $100$ digits in base $10$ representation. An element of $S$ is called [i]atom[/i] if it is not divisible by the sum of any two (not necessarily distinct) elements of $S$. If $S$ contains at most $10$ atoms, at most how many elements can $S$ have?
2003 CentroAmerican, 5
A square board with $8\text{cm}$ sides is divided into $64$ squares square with each side $1\text{cm}$. Each box can be painted white or black. Find the total number of ways to colour the board so that each square of side $2\text{cm}$ formed by four squares with a common vertex contains two white and two black squares.
2010 Contests, 3
Each of the small squares of a $50\times 50$ table is coloured in red or blue. Initially all squares are red. A [i]step[/i] means changing the colour of all squares on a row or on a column.
a) Prove that there exists no sequence of steps, such that at the end there are exactly $2011$ blue squares.
b) Describe a sequence of steps, such that at the end exactly $2010$ squares are blue.
[i]Adriana & Lucian Dragomir[/i]
2013 Saint Petersburg Mathematical Olympiad, 3
On a circle there are some black and white points (there are at least $12$ points). Each point has $10$ neighbors ($5$ left and $5$ right neighboring points), $5$ being black and $5$ white. Prove that the number of points on the circle is divisible by $4$.
2009 Miklós Schweitzer, 3
Prove that there exist positive constants $ c$ and $ n_0$ with the following property. If $ A$ is a finite set of integers, $ |A| \equal{} n > n_0$, then
\[ |A \minus{} A| \minus{} |A \plus{} A| \leq n^2 \minus{} c n^{8/5}.\]