Found problems: 1800
2011 Iran MO (2nd Round), 1
We have a line and $1390$ points around it such that the distance of each point to the line is less than $1$ centimeters and the distance between any two points is more than $2$ centimeters. prove that there are two points such that their distance is at least $10$ meters ($1000$ centimeters).
1989 Romania Team Selection Test, 3
Find all pair $(m,n)$ of integer ($m >1,n \geq 3$) with the following property:If an $n$-gon can be partitioned into $m$ isoceles triangles,then the $n$-gon has two congruent sides.
2011 China Team Selection Test, 3
For a given integer $n\ge 2$, let $a_0,a_1,\ldots ,a_n$ be integers satisfying $0=a_0<a_1<\ldots <a_n=2n-1$. Find the smallest possible number of elements in the set $\{ a_i+a_j \mid 0\le i \le j \le n \}$.
2015 China Team Selection Test, 3
Fix positive integers $k,n$. A candy vending machine has many different colours of candy, where there are $2n$ candies of each colour. A couple of kids each buys from the vending machine $2$ candies of different colours. Given that for any $k+1$ kids there are two kids who have at least one colour of candy in common, find the maximum number of kids.
2007 Junior Balkan Team Selection Tests - Romania, 3
Consider a $n$x$n$ table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a $2$x$2$ square which contains an odd number of unit squares white colored.
2002 Baltic Way, 8
Let $P$ be a set of $n\ge 3$ points in the plane, no three of which are on a line. How many possibilities are there to choose a set $T$ of $\binom{n-1}{2}$ triangles, whose vertices are all in $P$, such that each triangle in $T$ has a side that is not a side of any other triangle in $T$?
2010 IberoAmerican, 3
Around a circular table sit $12$ people, and on the table there are $28$ vases. Two people can see each other, if and only if there is no vase lined with them. Prove that there are at least two people who can be seen.
2005 Junior Balkan Team Selection Tests - Romania, 13
The positive integers from 1 to $n^2$ are placed arbitrarily on the $n^2$ squares of a $n\times n$ chessboard. Two squares are called [i]adjacent[/i] if they have a common side. Show that two opposite corner squares can be joined by a path of $2n-1$ adjacent squares so that the sum of the numbers placed on them is at least $\left\lfloor \frac{n^3} 2 \right\rfloor + n^2 - n + 1$.
[i]Radu Gologan[/i]
2006 Iran Team Selection Test, 2
Suppose $n$ coins are available that their mass is unknown. We have a pair of balances and every time we can choose an even number of coins and put half of them on one side of the balance and put another half on the other side, therefore a [i]comparison[/i] will be done. Our aim is determining that the mass of all coins is equal or not. Show that at least $n-1$ [i]comparisons[/i] are required.
2015 China National Olympiad, 3
Let $n \geq 5$ be a positive integer and let $A$ and $B$ be sets of integers satisfying the following conditions:
i) $|A| = n$, $|B| = m$ and $A$ is a subset of $B$
ii) For any distinct $x,y \in B$, $x+y \in B$ iff $x,y \in A$
Determine the minimum value of $m$.
2014 ELMO Shortlist, 6
Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$.
Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$.
Find a closed form for $a_n$.
[i]Proposed by Bobby Shen[/i]
1985 IMO Longlists, 32
A collection of $2n$ letters contains $2$ each of $n$ different letters. The collection is partitioned into $n$ pairs, each pair containing $2$ letters, which may be the same or different. Denote the number of distinct partitions by $u_n$. (Partitions differing in the order of the pairs in the partition or in the order of the two letters in the pairs are not considered distinct.) Prove that $u_{n+1}=(n+1)u_n - \frac{n(n-1)}{2} u_{n-2}.$
[i]Similar Problem :[/i]
A pack of $2n$ cards contains $n$ pairs of $2$ identical cards. It is shuffled and $2$ cards are dealt to each of $n$ different players. Let $p_n$ be the probability that every one of the $n$ players is dealt two identical cards. Prove that $\frac{1}{p_{n+1}}=\frac{n+1}{p_n} + \frac{n(n-1)}{2p_{n-2}}.$
1995 Irish Math Olympiad, 1
There are $ n^2$ students in a class. Each week all the students participate in a table quiz. Their teacher arranges them into $ n$ teams of $ n$ players each. For as many weeks as possible, this arrangement is done in such a way that any pair of students who were members of the same team one week are not in the same team in subsequent weeks. Prove that after at most $ n\plus{}2$ weeks, it is necessary for some pair of students to have been members of the same team in at least two different weeks.
2014 ELMO Shortlist, 4
Let $r$ and $b$ be positive integers. The game of [i]Monis[/i], a variant of Tetris, consists of a single column of red and blue blocks. If two blocks of the same color ever touch each other, they both vanish immediately. A red block falls onto the top of the column exactly once every $r$ years, while a blue block falls exactly once every $b$ years.
(a) Suppose that $r$ and $b$ are odd, and moreover the cycles are offset in such a way that no two blocks ever fall at exactly the same time. Consider a period of $rb$ years in which the column is initially empty. Determine, in terms of $r$ and $b$, the number of blocks in the column at the end.
(b) Now suppose $r$ and $b$ are relatively prime and $r+b$ is odd. At time $t=0$, the column is initially empty. Suppose a red block falls at times $t = r, 2r, \dots, (b-1)r$ years, while a blue block falls at times $t = b, 2b, \dots, (r-1)b$ years. Prove that at time $t=rb$, the number of blocks in the column is $\left\lvert 1+2(r-1)(b+r)-8S \right\rvert$, where \[ S = \left\lfloor \frac{2r}{r+b} \right\rfloor
+ \left\lfloor \frac{4r}{r+b} \right\rfloor
+ ...
+ \left\lfloor \frac{(r+b-1)r}{r+b} \right\rfloor
. \]
[i]Proposed by Sammy Luo[/i]
1979 Miklós Schweitzer, 7
Let $ T$ be a triangulation of an $ n$-dimensional sphere, and to each vertex of $ T$ let us assign a nonzero vector of a linear space $ V$. Show that if $ T$ has an $ n$-dimensional simplex such that the vectors assigned to the vertices of this simplex are linearly independent, then another such simplex must also exist.
[i]L. Lovasz[/i]
2002 Tournament Of Towns, 1
There are $2002$ employees in a bank. All the employees came to celebrate the bank's jubilee and were seated around one round table. It is known that the difference in salaries of any two adjacent employees is $2$ or $3$ dollars. Find the maximal difference in salaries of two employees, if it is known all salaries are different.
1988 IberoAmerican, 6
Consider all sets of $n$ distinct positive integers, no three of which form an arithmetic progression. Prove that among all such sets there is one which has the largest sum of the reciprocals of its elements.
2013 Iran Team Selection Test, 4
$m$ and $n$ are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows:
Starting from one of the hexagons, the Donkey moves $m$ cells in one of the $6$ directions, then it turns $60$ degrees clockwise and after that moves $n$ cells in this new direction until it reaches it's final cell.
At most how many cells are in the Philosopher's chessboard such that one cannot go from anyone of them to the other with a finite number of movements of the Donkey?
[i]Proposed by Shayan Dashmiz[/i]
2012 Tuymaada Olympiad, 1
Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy?
[i]Proposed by A. Golovanov[/i]
2013 All-Russian Olympiad, 4
On a $55\times 55$ square grid, $500$ unit squares were cut out as well as $400$ L-shaped pieces consisting of 3 unit squares (each piece can be oriented in any way) [refer to the figure]. Prove that at least two of the cut out pieces bordered each other before they were cut out.
[asy]size(2.013cm);
draw ((0,0)--(0,1));
draw ((0,0)--(1,0));
draw ((0,1)--(.5,1));
draw ((.5,1)--(.5,0));
draw ((0,.5)--(1,.5));
draw ((1,.5)--(1,0));
draw ((1,.5)--(1,0));
[/asy]
2012 ELMO Shortlist, 5
Form the infinite graph $A$ by taking the set of primes $p$ congruent to $1\pmod{4}$, and connecting $p$ and $q$ if they are quadratic residues modulo each other. Do the same for a graph $B$ with the primes $1\pmod{8}$. Show $A$ and $B$ are isomorphic to each other.
[i]Linus Hamilton.[/i]
2012 Turkey MO (2nd round), 5
Let $P$ be the set of all $2012$ tuples $(x_1, x_2, \dots, x_{2012})$, where $x_i \in \{1,2,\dots 20\}$ for each $1\leq i \leq 2012$. The set $A \subset P$ is said to be decreasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in A$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \leq x_i (1\leq i \leq 2012)$ also belongs to $A$. The set $B \subset P$ is said to be increasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in B$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \geq x_i (1\leq i \leq 2012)$ also belongs to $B$. Find the maximum possible value of $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$, where $A$ and $B$ are nonempty decreasing and increasing sets ($\mid \cdot \mid$ denotes the number of elements of the set).
2011 All-Russian Olympiad, 2
There are more than $n^2$ stones on the table. Peter and Vasya play a game, Peter starts. Each turn, a player can take any prime number less than $n$ stones, or any multiple of $n$ stones, or $1$ stone. Prove that Peter always can take the last stone (regardless of Vasya's strategy).
[i]S Berlov[/i]
2007 France Team Selection Test, 1
Do there exist $5$ points in the space, such that for all $n\in\{1,2,\ldots,10\}$ there exist two of them at distance between them $n$?
2003 Iran MO (2nd round), 3
We have a chessboard and we call a $1\times1$ square a room. A robot is standing on one arbitrary vertex of the rooms. The robot starts to move and in every one movement, he moves one side of a room. This robot has $2$ memories $A,B$. At first, the values of $A,B$ are $0$. In each movement, if he goes up, $1$ unit is added to $A$, and if he goes down, $1$ unit is waned from $A$, and if he goes right, the value of $A$ is added to $B$, and if he goes left, the value of $A$ is waned from $B$. Suppose that the robot has traversed a traverse (!) which hasn’t intersected itself and finally, he has come back to its initial vertex. If $v(B)$ is the value of $B$ in the last of the traverse, prove that in this traverse, the interior surface of the shape that the robot has moved on its circumference is equal to $|v(B)|$.