Found problems: 1488
2008 Tournament Of Towns, 1
$100$ Queens are placed on a $100 \times 100$ chessboard so that no two attack each other. Prove that each of four $50 \times 50$ corners of the board contains at least one Queen.
2011 Tuymaada Olympiad, 1
Each real number greater than 1 is colored red or blue with both colors being used. Prove that there exist real numbers $a$ and $b$ such that the numbers $a+\frac1b$ and $b+\frac1a$ are different colors.
2012 Baltic Way, 10
Two players $A$ and $B$ play the following game. Before the game starts, $A$ chooses 1000 not necessarily different odd primes, and then $B$ chooses half of them and writes them on a blackboard. In each turn a player chooses a positive integer $n$, erases some primes $p_1$, $p_2$, $\dots$, $p_n$ from the blackboard and writes all the prime factors of $p_1 p_2 \dotsm p_n - 2$ instead (if a prime occurs several times in the prime factorization of $p_1 p_2 \dotsm p_n - 2$, it is written as many times as it occurs). Player $A$ starts, and the player whose move leaves the blackboard empty loses the game. Prove that one of the two players has a winning strategy and determine who.
Remark: Since 1 has no prime factors, erasing a single 3 is a legal move.
2010 Kurschak Competition, 1
We have $n$ keys, each of them belonging to exactly one of $n$ locked chests. Our goal is to decide which key opens which chest. In one try we may choose a key and a chest, and check whether the chest can be opened with the key. Find the minimal number $p(n)$ with the property that using $p(n)$ tries, we can surely discover which key belongs to which chest.
2018 IFYM, Sozopol, 7
$n$ points were chosen on a circle. Two players are playing the following game: On every move a point is chosen and it is connected with an edge to an adjacent point or with the center of the circle. The winner is the player, after whose move each point can be reached by any other (including the center) by moving on the constructed edges. Find who of the two players has a winning strategy.
2001 Tournament Of Towns, 5
[b](a)[/b] One black and one white pawn are placed on a chessboard. You may move the pawns in turn to the neighbouring empty squares of the chessboard using vertical and horizontal moves. Can you arrange the moves so that every possible position of the two pawns will appear on the chessboard exactly once?
[b](b)[/b] Same question, but you don’t have to move the pawns in turn.
2018 Latvia Baltic Way TST, P6
Let $ABCD$ be a rectangle consisting of unit squares. All vertices of these unit squares inside the rectangle and on its sides have been colored in four colors. Additionally, it is known that:
[list]
[*] every vertex that lies on the side $AB$ has been colored in either the $1.$ or $2.$ color;
[*] every vertex that lies on the side $BC$ has been colored in either the $2.$ or $3.$ color;
[*] every vertex that lies on the side $CD$ has been colored in either the $3.$ or $4.$ color;
[*] every vertex that lies on the side $DA$ has been colored in either the $4.$ or $1.$ color;
[*] no two neighboring vertices have been colored in $1.$ and $3.$ color;
[*] no two neighboring vertices have been colored in $2.$ and $4.$ color.
[/list]
Notice that the constraints imply that vertex $A$ has been colored in $1.$ color etc. Prove that there exists a unit square that has all vertices in different colors (in other words it has one vertex of each color).
1985 USAMO, 3
Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$. Determine the maximum value of the sum of the six distances.
1997 Romania Team Selection Test, 2
Find the number of sets $A$ containing $9$ positive integers with the following property: for any positive integer $n\le 500$, there exists a subset $B\subset A$ such that $\sum_{b\in B}{b}=n$.
[i]Bogdan Enescu & Dan Ismailescu[/i]
2008 Turkey Team Selection Test, 2
A graph has $ 30$ vertices, $ 105$ edges and $ 4822$ unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph.
1972 IMO Longlists, 24
The diagonals of a convex $18$-gon are colored in $5$ different colors, each color appearing on an equal number of diagonals. The diagonals of one color are numbered $1, 2,\cdots$. One randomly chooses one-fifth of all the diagonals. Find the number of possibilities for which among the chosen diagonals there exist exactly $n$ pairs of diagonals of the same color and with fixed indices $i, j$.
2010 Finnish National High School Mathematics Competition, 4
In a football season, even number $n$ of teams plays a simple series, i.e. each team plays once against each other team. Show that ona can group the series into $n-1$ rounds such that in every round every team plays exactly one match.
2001 India IMO Training Camp, 3
Each vertex of an $m\times n$ grid is colored blue, green or red in such a way that all the boundary vertices are red. We say that a unit square of the grid is properly colored if:
$(i)$ all the three colors occur at the vertices of the square, and
$(ii)$ one side of the square has the endpoints of the same color.
Show that the number of properly colored squares is even.
2009 Ukraine National Mathematical Olympiad, 2
Let $M = \{1, 2, 3, 4, 6, 8,12,16, 24, 48\} .$ Find out which of four-element subsets of $M$ are more: those with product of all elements greater than $2009$ or those with product of all elements less than $2009.$
2022 Israel TST, 3
A class has 30 students. To celebrate 'Tu BiShvat' each student chose some dried fruits out of $n$ different kinds. Say two students are friends if they both chose from the same type of fruit. Find the minimal $n$ so that it is possible that each student has exactly \(6\) friends.
1997 Belarusian National Olympiad, 3
Does there exist an infinite set $ M$ of straight lines on the coordinate plane such that
(i) no two lines are parallel, and
(ii) for any integer point there is a line from $ M$ containing it?
2013 India Regional Mathematical Olympiad, 6
For a natural number $n$, let $T(n)$ denote the number of ways we can place $n$ objects of weights $1,2,\cdots, n$ on a balance such that the sum of the weights in each pan is the same. Prove that $T(100) > T(99)$.
1990 IMO Longlists, 99
Given a $10 \times 10$ chessboard colored as black-and-white alternately. Prove that for any $46$ unit squares without common edges, there are at least $30$ unit squares with the same color.
2001 IberoAmerican, 2
In a board of $2000\times2001$ squares with integer coordinates $(x,y)$, $0\leq{x}\leq1999$ and $0\leq{y}\leq2000$. A ship in the table moves in the following way: before a move, the ship is in position $(x,y)$ and has a velocity of $(h,v)$ where $x,y,h,v$ are integers. The ship chooses new velocity $(h^\prime,v^\prime)$ such that $h^\prime-h,v^\prime-v\in\{-1,0,1\}$. The new position of the ship will be $(x^\prime,y^\prime)$ where $x^\prime$ is the remainder of the division of $x+h^\prime$ by $2000$ and $y^\prime$ is the remainder of the division of $y+v^\prime$ by $2001$.
There are two ships on the board: The Martian ship and the Human trying to capture it. Initially each ship is in a different square and has velocity $(0,0)$. The Human is the first to move; thereafter they continue moving alternatively.
Is there a strategy for the Human to capture the Martian, independent of the initial positions and the Martian’s moves?
[i]Note[/i]: The Human catches the Martian ship by reaching the same position as the Martian ship after the same move.
1986 China National Olympiad, 6
Suppose that each point on the plane is colored either white or black. Show that there exists an equilateral triangle with the side length equal to $1$ or $\sqrt{3}$ whose three vertices are in the same color.
2005 Cono Sur Olympiad, 3
On the cartesian plane we draw circunferences of radii 1/20 centred in each lattice point. Show that any circunference of radii 100 in the cartesian plane intersect at least one of the small circunferences.
2010 China Team Selection Test, 3
An (unordered) partition $P$ of a positive integer $n$ is an $n$-tuple of nonnegative integers $P=(x_1,x_2,\cdots,x_n)$ such that $\sum_{k=1}^n kx_k=n$. For positive integer $m\leq n$, and a partition $Q=(y_1,y_2,\cdots,y_m)$ of $m$, $Q$ is called compatible to $P$ if $y_i\leq x_i$ for $i=1,2,\cdots,m$. Let $S(n)$ be the number of partitions $P$ of $n$ such that for each odd $m<n$, $m$ has exactly one partition compatible to $P$ and for each even $m<n$, $m$ has exactly two partitions compatible to $P$. Find $S(2010)$.
1985 IMO Longlists, 50
From each of the vertices of a regular $n$-gon a car starts to move with constant speed along the perimeter of the $n$-gon in the same direction. Prove that if all the cars end up at a vertex $A$ at the same time, then they never again meet at any other vertex of the $n$-gon. Can they meet again at $A \ ?$
2003 China Team Selection Test, 2
Suppose $A=\{1,2,\dots,2002\}$ and $M=\{1001,2003,3005\}$. $B$ is an non-empty subset of $A$. $B$ is called a $M$-free set if the sum of any two numbers in $B$ does not belong to $M$. If $A=A_1\cup A_2$, $A_1\cap A_2=\emptyset$ and $A_1,A_2$ are $M$-free sets, we call the ordered pair $(A_1,A_2)$ a $M$-partition of $A$. Find the number of $M$-partitions of $A$.
1996 USAMO, 4
An $n$-term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a [i]binary sequence of length [/i]$n$. Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.