Found problems: 1800
2003 India National Olympiad, 6
Each lottery ticket has a 9-digit numbers, which uses only the digits $1$, $2$, $3$. Each ticket is colored [color=red]red[/color],[color=blue] blue [/color]or [color=green]green[/color]. If two tickets have numbers which differ in all nine places, then the tickets have different colors. Ticket $122222222$ is red, and ticket $222222222$ is [color=green]green.[/color] What color is ticket $123123123$ ?
2007 Tournament Of Towns, 1
Pictures are taken of $100$ adults and $100$ children, with one adult and one child in each, the adult being the taller of the two. Each picture is reduced to $\frac 1k$ of its original size, where $k$ is a positive integer which may vary from picture to picture. Prove that it is possible to have the reduced image of each adult taller than the reduced image of every child.
2008 Korea - Final Round, 6
There is $n\times n$ chessboard. Each square has a number between $0$ and $k$. There is a button for each row and column, which increases the number of $n$ numbers of the row or column the button represents(if the number of the square is $k$, then it becomes $0$). If certain button is pressed, call it 'operation.'
And we have a chessboard which is filled with 0(for all squares). After some 'operation's, the numbers of squares are different now. Prove that we can make all of the number $0$ within $kn$ 'operation's.
2013 Tuymaada Olympiad, 4
The vertices of a connected graph cannot be coloured with less than $n+1$ colours (so that adjacent vertices have different colours).
Prove that $\dfrac{n(n-1)}{2}$ edges can be removed from the graph so that it remains connected.
[i]V. Dolnikov[/i]
[b]EDIT.[/b] It is confirmed by the official solution that the graph is tacitly assumed to be [b]finite[/b].
2002 China Western Mathematical Olympiad, 4
Let $ n$ be a positive integer, let the sets $ A_{1},A_{2},\cdots,A_{n \plus{} 1}$ be non-empty subsets of the set $ \{1,2,\cdots,n\}.$ prove that there exist two disjoint non-empty subsets of the set $ \{1,2,\cdots,n \plus{} 1\}$: $ \{i_{1},i_{2},\cdots,i_{k}\}$ and $ \{j_{1},j_{2},\cdots,j_{m}\}$ such that $ A_{i_{1}}\cup A_{i_{2}}\cup\cdots\cup A_{i_{k}} \equal{} A_{j_{1}}\cup A_{j_{2}}\cup\cdots\cup A_{j_{m}}$.
2010 Balkan MO Shortlist, C3
A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$.
Prove that $S$ can be covered by a strip of width $2$.
2005 USA Team Selection Test, 5
Find all finite sets $S$ of points in the plane with the following property: for any three distinct points $A,B,$ and $C$ in $S,$ there is a fourth point $D$ in $S$ such that $A,B,C,$ and $D$ are the vertices of a parallelogram (in some order).
2000 Taiwan National Olympiad, 2
Let $n$ be a positive integer and $A=\{ 1,2,\ldots ,n\}$. A subset of $A$ is said to be connected if it consists of one element or several consecutive elements. Determine the maximum $k$ for which there exist $k$ distinct subsets of $A$ such that the intersection of any two of them is connected.
2012 Cono Sur Olympiad, 1
1. Around a circumference are written $2012$ number, each of with is equal to $1$ or $-1$. If there are not $10$ consecutive numbers that sum $0$, find all possible values of the sum of the $2012$ numbers.
2002 Romania Team Selection Test, 1
Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$. Find the least number $k$ such that $s(M,N)\le k$, for all points $M,N$.
[i]Dinu Șerbănescu[/i]
2001 China National Olympiad, 3
Let $P$ be a regular $n$-gon $A_1A_2\ldots A_n$. Find all positive integers $n$ such that for each permutation $\sigma (1),\sigma (2),\ldots ,\sigma (n)$ there exists $1\le i,j,k\le n$ such that the triangles $A_{i}A_{j}A_{k}$ and $A_{\sigma (i)}A_{\sigma (j)}A_{\sigma (k)}$ are both acute, both right or both obtuse.
2010 ELMO Problems, 2
2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010 are chosen. On the red slip of paper, each MOPper writes the remainder when the product of A and his or her number is divided by 2011. On the blue slip of paper, he or she writes the remainder when the product of B and his or her number is divided by 2011. The MOPpers may then perform either of the following two operations:
[list]
[*] Each MOPper gives his or her red slip to the MOPper whose number is written on his or her blue slip.
[*] Each MOPper gives his or her blue slip to the MOPper whose number is written on his or her red slip.[/list]
Show that it is always possible to perform some number of these operations such that each MOPper is holding a red slip with his or her number written on it.
[i]Brian Hamrick.[/i]
1985 IMO Longlists, 45
Two persons, $X$ and $Y$ , play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. For each player determine the probability of winning a match within $5$ games. Determine the probabilities of winning in an unlimited number of games. If $X$ bets $1$, how much must $Y$ bet for the game to be fair ?
2003 Tournament Of Towns, 1
$2003$ dollars are placed into $N$ purses, and the purses are placed into $M$ pockets. It is known that $N$ is greater than the number of dollars in any pocket. Is it true that there is a purse with less than $M$ dollars in it?
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?
2014 National Olympiad First Round, 24
If the integers $1,2,\dots,n$ can be divided into two sets such that each of the two sets does not contain the arithmetic mean of its any two elements, what is the largest possible value of $n$?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 9
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2006 Polish MO Finals, 1
Given a triplet we perform on it the following operation. We choose two numbers among them and change them into their sum and product, left number stays unchanged. Can we, starting from triplet $(3,4,5)$ and performing above operation, obtain again a triplet of numbers which are lengths of right triangle?
1998 Baltic Way, 16
Is it possible to cover a $13\times 13$ chessboard with forty-two pieces of dimensions $4\times 1$ such that only the central square of the chessboard remains uncovered?
1988 Romania Team Selection Test, 8
The positive integer $n$ is given and for all positive integers $k$, $1\leq k\leq n$, denote by $a_{kn}$ the number of all ordered sequences $(i_1,i_2,\ldots,i_k)$ of positive integers which verify the following two conditions:
a) $1\leq i_1<i_2< \cdots i_k \leq n$;
b) $i_{r+1}-i_r \equiv 1 \pmod 2$, for all $r \in\{1,2,\ldots,k-1\}$.
Compute the number $a(n) = \sum\limits_{k=1}^n a_{kn}$.
[i]Ioan Tomescu[/i]
2011 Pre-Preparation Course Examination, 5
[b]a)[/b] Prove that if $G$ is $2$-connected, then it has a cycle with the length at least $\min\{n(G),2\delta(G)\}$. (10 points)
[b]b)[/b] Prove that every $2k$-regular graph with $4k+1$ vertices has a hamiltonian cycle. (10 points)
2009 China Team Selection Test, 2
Find all the pairs of integers $ (a,b)$ satisfying $ ab(a \minus{} b)\not \equal{} 0$ such that there exists a subset $ Z_{0}$ of set of integers $ Z,$ for any integer $ n$, exactly one among three integers $ n,n \plus{} a,n \plus{} b$ belongs to $ Z_{0}$.
2008 IMAR Test, 1
An array $ n\times n$ is given, consisting of $ n^2$ unit squares. A [i]pawn[/i] is placed arbitrarily on a unit square. The pawn can move from a square of the $ k$-th column to any square of the $ k$-th row. Show that there exists a sequence of $ n^2$ moves of the pawn so that all the unit squares of the array are visited once, the pawn returning to its original position.
[b]Dinu Serbanescu[/b]
1973 IMO Longlists, 6
Let $P_i (x_i, y_i)$ (with $i = 1, 2, 3, 4, 5$) be five points with integer coordinates, no three collinear. Show that among all triangles with vertices at these points, at least three have integer areas.
2010 Contests, 3
A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$.
Prove that $S$ can be covered by a strip of width $2$.
2007 Iran Team Selection Test, 2
Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\frac{\pi}2]$. Find the largest value of the sum of the $\binom n2$ angles between line.
[i]By Aliakbar Daemi[/i]