Found problems: 1488
1990 IMO Longlists, 64
Given an $m$-element set $M$ and a $k$-element subset $K \subset M$. We call a function $f: K \to M$ has "path", if there exists an element $x_0 \in K$ such that $f(x_0) = x_0$, or there exists a chain $x_0, x_1, \ldots, x_j = x_0 \in K$ such that $_xi = f(x_{i-1})$ for $i = 1, 2, \ldots, j$. Find the number of functions $f: K \to M$ which have path.
2004 Rioplatense Mathematical Olympiad, Level 3, 2
A collection of cardboard circles, each with a diameter of at most $1$, lie on a $5\times 8$ table without overlapping or overhanging the edge of the table. A cardboard circle of diameter $2$ is added to the collection. Prove that this new collection of cardboard circles can be placed on a $7\times 7$ table without overlapping or overhanging the edge.
1992 China Team Selection Test, 2
A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.
2012 Finnish National High School Mathematics Competition, 4
Let $k,n\in\mathbb{N},0<k<n.$ Prove that \[\sum_{j=1}^k\binom{n}{j}=\binom{n}{1}+ \binom{n}{2}+\ldots + \binom{n}{k}\leq n^k.\]
2005 Polish MO Finals, 3
In a matrix $2n \times 2n$, $n \in N$, are $4n^2$ real numbers with a sum equal zero. The absolute value of each of these numbers is not greater than $1$. Prove that the absolute value of a sum of all the numbers from one column or a row doesn't exceed $n$.
2014 Postal Coaching, 4
Let $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$ be two partitions of a set $M$ such that $|A_j\cup B_k|\ge n$ for any $j,k\in\{1,2,\ldots,n\}$. Prove that $|M|\ge n^2/2$.
2004 India IMO Training Camp, 3
Two runners start running along a circular track of unit length from the same starting point and int he same sense, with constant speeds $v_1$ and $v_2$ respectively, where $v_1$ and $v_2$ are two distinct relatively prime natural numbers. They continue running till they simultneously reach the starting point. Prove that
(a) at any given time $t$, at least one of the runners is at a distance not more than $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units from the starting point.
(b) there is a time $t$ such that both the runners are at least $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units away from the starting point. (All disstances are measured along the track). $[x]$ is the greatest integer function.
1998 All-Russian Olympiad, 5
Initially the numbers $19$ and $98$ are written on a board. Every minute, each of the two numbers is either squared or increased by $1$. Is it possible to obtain two equal numbers at some time?
1994 All-Russian Olympiad, 8
Players $ A,B$ alternately move a knight on a $ 1994\times 1994$ chessboard. Player $ A$ makes only horizontal moves, i.e. such that the knight is moved to a neighboring row, while $ B$ makes only vertical moves. Initally player $ A$ places the knight to an arbitrary square and makes the first move. The knight cannot be moved to a square that was already visited during the game. A player who cannot make a move loses. Prove that player $ A$ has a winning strategy.
1987 IMO Longlists, 37
Five distinct numbers are drawn successively and at random from the set $\{1, \cdots , n\}$. Show that the probability of a draw in which the first three numbers as well as all five numbers can be arranged to form an arithmetic progression is greater than $\frac{6}{(n-2)^3}$
2014 India IMO Training Camp, 3
In how many ways rooks can be placed on a $8$ by $8$ chess board such that every row and every column has at least one rook?
(Any number of rooks are available,each square can have at most one rook and there is no relation of attacking between them)
1995 Greece National Olympiad, 4
Given are the lines $l_1,l_2,\ldots ,l_k$ in the plane, no two of which are parallel and no three of which are concurrent. For which $k$ can one label the intersection points of these lines by $1, 2,\ldots , k-1$ so that in each of the given lines all the labels appear exactly once?
2004 239 Open Mathematical Olympiad, 2
Do there exist such a triangle $T$, that for any coloring of a plane in two colors one may found a triangle $T'$, equal to $T$, such that all vertices of $T'$ have the same color.
[b]
proposed by S. Berlov[/b]
2007 South East Mathematical Olympiad, 4
A sequence of positive integers with $n$ terms satisfies $\sum_{i=1}^{n} a_i=2007$. Find the least positive integer $n$ such that there exist some consecutive terms in the sequence with their sum equal to $30$.
2002 May Olympiad, 5
Let $x$ and $y$ be positive integers we have a table $x\times y$ where $(x + 1)(y + 1)$ points are red(the points are the vertices of the squares). Initially there is one ant in each red point, in a moment the ants walk by the lines of the table with same speed, each turn that an ant arrive in a red point the ant moves $90º$ to some direction.
Determine all values of $x$ and $y$ where is possible that the ants move indefinitely where can't be in any moment two ants in the same red point.
2007 Hungary-Israel Binational, 1
You have to organize a fair procedure to randomly select someone from $ n$ people so that every one of them would be chosen with the probability $ \frac{1}{n}$. You are allowed to choose two real numbers $ 0<p_1<1$ and $ 0<p_2<1$ and order two coins which satisfy the following requirement: the probability of tossing "heads" on the first coin $ p_1$ and the probability of tossing "heads" on the second coin is $ p_2$. Before starting the procedure, you are supposed to announce an upper bound on the total number of times that the two coins are going to be flipped altogether. Describe a procedure that achieves this goal under the given conditions.
1998 All-Russian Olympiad, 4
Let $k$ be a positive integer. Some of the $2k$-element subsets of a given set are marked. Suppose that for any subset of cardinality less than or equal to $(k+1)^2$ all the marked subsets contained in it (if any) have a common element. Show that all the marked subsets have a common element.
1986 Kurschak Competition, 1
Any two members of a club with $3n+1$ people plays ping-pong, tennis or chess with each other. Everyone has exactly $n$ partners who plays ping-pong, $n$ who play tennis and $n$ who play chess.
Prove that we can choose three members of the club who play three different games amongst each other.
1985 IMO Longlists, 61
Consider the set $A = \{0, 1, 2, \dots , 9 \}$ and let $(B_1,B_2, \dots , B_k)$ be a collection of nonempty subsets of $A$ such that $B_i \cap B_j$ has at most two elements for $i \neq j$. What is the maximal value of $k \ ?$
2013 ELMO Shortlist, 1
Let $n\ge2$ be a positive integer. The numbers $1,2,..., n^2$ are consecutively placed into squares of an $n\times n$, so the first row contains $1,2,...,n$ from left to right, the second row contains $n+1,n+2,...,2n$ from left to right, and so on. The [i]magic square value[/i] of a grid is defined to be the number of rows, columns, and main diagonals whose elements have an average value of $\frac{n^2 + 1}{2}$. Show that the magic-square value of the grid stays constant under the following two operations: (1) a permutation of the rows; and (2) a permutation of the columns. (The operations can be used multiple times, and in any order.)
[i]Proposed by Ray Li[/i]
1995 Polish MO Finals, 2
An urn contains $n$ balls labeled $1, 2, ... , n$. We draw the balls out one by one (without replacing them) until we obtain a ball whose number is divisible by $k$. Find all $k$ such that the expected number of balls removed is $k$.
2003 Tournament Of Towns, 5
Two players in turn play a game. First Player has cards with numbers $2, 4, \ldots, 2000$ while Second Player has cards with numbers $1, 3, \ldots, 2001$. In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card next to the first one. Player, who put the card with a larger number, scores 1 point. Then both cards are discarded. First Player starts. After $1000$ turns the game is over; First Player has used all his cards and Second Player used all but one. What are the maximal scores, that players could guarantee for themselves, no matter how the opponent would play?
1988 Polish MO Finals, 2
For a permutation $P = (p_1, p_2, ... , p_n)$ of $(1, 2, ... , n)$ define $X(P)$ as the number of $j$ such that $p_i < p_j$ for every $i < j$. What is the expected value of $X(P)$ if each permutation is equally likely?
1998 Taiwan National Olympiad, 6
In a group of $n\geq 4$ persons, every three who know each other have a common signal. Assume that these signals are not repeatad and that there are $m\geq 1$ signals in total. For any set of four persons in which there are three having a common signal, the fourth person has a common signal with at most one of them. Show that there three persons who have a common signal, such that the number of persons having no signal with anyone of them does not exceed $[n+3-\frac{18m}{n}]$.
2004 Baltic Way, 11
Given a table $m\times n$, in each cell of which a number $+1$ or $-1$ is written. It is known that initially exactly one $-1$ is in the table, all the other numbers being $+1$. During a move, it is allowed to chose any cell containing $-1$, replace this $-1$ by $0$, and simultaneously multiply all the numbers in the neighbouring cells by $-1$ (we say that two cells are neighbouring if they have a common side). Find all $(m,n)$ for which using such moves one can obtain the table containing zeros only, regardless of the cell in which the initial $-1$ stands.