Found problems: 1800
2020 Baltic Way, 6
Let $n>2$ be a given positive integer. There are $n$ guests at Georg's bachelor party and each guest is friends with at least one other guest. Georg organizes a party game among the guests. Each guest receives a jug of water such that there are no two guests with the same amount of water in their jugs. All guests now proceed simultaneously as follows. Every guest takes one cup for each of his friends at the party and distributes all the water from his jug evenly in the cups. He then passes a cup to each of his friends. Each guest having received a cup of water from each of his friends pours the water he has received into his jug. What is the smallest possible number of guests that do not have the same amount of water as they started with?
2009 All-Russian Olympiad, 6
Can be colored the positive integers with 2009 colors if we know that each color paints infinitive integers and that we can not find three numbers colored by three different colors for which the product of two numbers equal to the third one?
2008 Irish Math Olympiad, 4
How many sequences $ a_1,a_2,...,a{}_2{}_0{}_0{}_8$ are there such that each of the numbers $ 1,2,...,2008$ occurs once in the sequence, and $ i \in (a_1,a_2,...,a_i)$ for each $ i$ such that $ 2\le i \le2008$?
2009 Romania Team Selection Test, 2
Consider a matrix whose entries are integers. Adding a same integer to all entries on a same row, or on a same column, is called an operation. It is given that, for infinitely many positive integers $n$, one can obtain, through a finite number of operations, a matrix having all entries divisible by $n$. Prove that, through a finite number of operations, one can obtain the null matrix.
2014 Contests, 2
$3m$ balls numbered $1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m$ are distributed into $8$ boxes so that any two boxes contain identical balls. Find the minimal possible value of $m$.
2024 Dutch IMO TST, 1
On a $2023 \times 2023$ board, there are beetles on some of the cells, with at most one beetle per cell. After one minute, each beetle moves a cell to the right or to the left or to the top or to the bottom. After each further minute, the beetles continue to move to adjacent fields, but they always make a $90^\circ$ turn, i.e. when a beetle just moved to the right or to the left, it now moves to the top or to the bottom, and vice versa. What is the minimal number of beetles on the board such that no matter where they start and how they move (according to the rules), at some point two beetles will end up in the cell?
2005 All-Russian Olympiad Regional Round, 9.2
9.2 Given 19 cards. Is it possible to write a nonzero digit on each card in such a way that you can compose from these cards an unique 19-digits number, which is divisible by 11?
([i]R. Zhenodarov, I. Bogdanov[/i])
2011 South East Mathematical Olympiad, 4
12 points are located on a clock with the sme distance , numbers $1,2,3 , ... 12$ are marked on each point in clockwise order . Use 4 kinds of colors (red,yellow,blue,green) to colour the the points , each kind of color has 3 points . N ow , use these 12 points as the vertex of convex quadrilateral to construct $n$ convex quadrilaterals . They satisfies the following conditions:
(1). the colours of vertex of every convex quadrilateral are different from each other .
(2). for every 3 quadrilaterals among them , there exists a colour such that : the numbers on the 3 points painted into this colour are different from each other .
Find the maximum $n$ .
2002 Tournament Of Towns, 4
In how many ways can we place the numbers from $1$ to $100$ in a $2\times 50$ rectangle (divided into $100$ unit squares) so that any two consecutive numbers are always placed in squares with a common side?
2024 All-Russian Olympiad, 7
Let $x_1$ and $x_2$ be positive integers. On a straight line, $y_1$ white segments and $y_2$ black segments are given, with $y_1 \ge x_1$ and $y_2 \ge x_2$. Suppose that no two segments of the same colour intersect (and do not have common ends). Moreover, suppose that for any choice of $x_1$ white segments and $x_2$ black segments, some pair of selected segments will intersect. Prove that $(y_1-x_1)(y_2-x_2)<x_1x_2$.
[i]Proposed by G. Chelnokov[/i]
1989 Turkey Team Selection Test, 4
There is a stone on each square of $n\times n$ chessboard. We gather $n^2$ stones and distribute them to the squares (again each square contains one stone) such that any two adjacent stones are again adjacent. Find all distributions such that at least one stone at the corners remains at its initial square. (Two squares are adjacent if they share a common edge.)
2005 Baltic Way, 10
Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$, there exist 3 numbers $a$, $b$, $c$ among them satisfying $abc=m$.
2004 Germany Team Selection Test, 3
Let $n \geq 2$ be a natural number, and let $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ be a permutation of $\left(1;\;2;\;...;\;n\right)$. For any integer $k$ with $1 \leq k \leq n$, we place $a_k$ raisins on the position $k$ of the real number axis. [The real number axis is the $x$-axis of a Cartesian coordinate system.]
Now, we place three children A, B, C on the positions $x_A$, $x_B$, $x_C$, each of the numbers $x_A$, $x_B$, $x_C$ being an element of $\left\{1;\;2;\;...;\;n\right\}$. [It is not forbidden to place different children on the same place!]
For any $k$, the $a_k$ raisins placed on the position $k$ are equally handed out to those children whose positions are next to $k$. [So, if there is only one child lying next to $k$, then he gets the raisin. If there are two children lying next to $k$ (either both on the same position or symmetric with respect to $k$), then each of them gets one half of the raisin. Etc..]
After all raisins are distributed, a child is unhappy if he could have received more raisins than he actually has received if he had moved to another place (while the other children would rest on their places).
For which $n$ does there exist a configuration $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ and numbers $x_A$, $x_B$, $x_C$, such that all three children are happy?
2008 Tournament Of Towns, 1
A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant ) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between area of white rectangle and area of black rectangle does not exceed $2.$ Determine the maximal ratio between area of white and black part of the board. White (black) part of the board is the total sum of area of all white (black) rectangles.
2005 Iran MO (3rd Round), 4
Suppose we have some proteins that each protein is a sequence of 7 "AMINO-ACIDS" $A,\ B,\ C,\ H,\ F,\ N$. For example $AFHNNNHAFFC$ is a protein. There are some steps that in each step an amino-acid will change to another one. For example with the step $NA\rightarrow N$ the protein $BANANA$ will cahnge to $BANNA$("in Persian means workman"). We have a set of allowed steps that each protein can change with these steps. For example with the
set of steps:
$\\ 1)\ AA\longrightarrow A\\ 2)\ AB\longrightarrow BA\\ 3)\ A\longrightarrow \mbox{null}$
Protein $ABBAABA$ will change like this:
$\\ ABB\underline{AA}BA\\ \underline{AB}BABA\\ B\underline{AB}ABA\\ BB\underline{AA}BA\\ BB\underline{AB}A\\ BBB\underline{AA}\\ BBB\underline{A}\\ BBB$
You see after finite steps this protein will finish it steps.
Set of allowed steps that for them there exist a protein that may have infinitely many steps is dangerous. Which of the following allowed sets are dangerous?
a) $NO\longrightarrow OONN$
b) $\left\{\begin{array}{c}HHCC\longrightarrow HCCH\\ CC\longrightarrow CH\end{array}\right.$
c) Design a set of allowed steps that change $\underbrace{AA\dots A}_{n}\longrightarrow\underbrace{BB\dots B}_{2^{n}}$
d) Design a set of allowed steps that change $\underbrace{A\dots A}_{n}\underbrace{B\dots B}_{m}\longrightarrow\underbrace{CC\dots C}_{mn}$
You see from $c$ and $d$ that we acn calculate the functions $F(n)=2^{n}$ and $G(M,N)=mn$ with these steps. Find some other calculatable functions with these steps. (It has some extra mark.)
1999 Turkey Team Selection Test, 2
Each of $A$, $B$, $C$, $D$, $E$, and $F$ knows a piece of gossip. They communicate by telephone via a central switchboard, which can connect only two of them at a time. During a conversation, each side tells the other everything he or she knows at that point. Determine the minimum number of calls for everyone to know all six pieces of gossip.
2009 Romania Team Selection Test, 1
We call Golomb ruler a ruler of length $l$, bearing $k+1\geq 2$ marks $0<a_1<\ldots <a_{k-1}<l$, such that the lengths that can be measured using marks on the ruler are consecutive integers starting with $1$, and each such length be measurable between just two of the gradations of the ruler. Find all Golomb rulers.
2005 Nordic, 3
There are $2005$ young people sitting around a large circular table. Of these, at most $668$ are boys. We say that a girl $G$ has a strong position, if, counting from $G$ in either direction, the number of girls is always strictly larger than the number of boys ($G$ is herself included in the count). Prove that there is always a girl in a strong position.
2012 Romania National Olympiad, 4
[color=darkred]On a table there are $k\ge 2$ piles having $n_1,n_2,\ldots,n_k$ pencils respectively. A [i]move[/i] consists in choosing two piles having $a$ and $b$ pencils respectively, $a\ge b$ and transferring $b$ pencils from the first pile to the second one. Find the necessary and sufficient condition for $n_1,n_2,\ldots,n_k$ , such that there exists a succession of moves through which all pencils are transferred to the same pile.[/color]
2010 Contests, 4
Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$
2009 Miklós Schweitzer, 1
On every card of a deck of cards a regular 17-gon is displayed with all sides and diagonals, and the vertices are numbered from 1 through 17. On every card all edges (sides and diagonals) are colored with a color 1,2,...,105 such that the following property holds: for every 15 vertices of the 17-gon the 105 edges connecting these vertices are colored with different colors on at least one of the cards. What is the minimum number of cards in the deck?
1996 All-Russian Olympiad, 2
On a coordinate plane are placed four counters, each of whose centers has integer coordinates. One can displace any counter by the vector joining the centers of two of the other counters. Prove that any two preselected counters can be made to coincide by a finite sequence of moves.
[i]Р. Sadykov[/i]
2008 Bosnia And Herzegovina - Regional Olympiad, 4
A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$, $ 2$, ...,$ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$. What is maximum value of minimum sum in column (with minimal sum)?
2011 China Team Selection Test, 2
Let $S$ be a set of $n$ points in the plane such that no four points are collinear. Let $\{d_1,d_2,\cdots ,d_k\}$ be the set of distances between pairs of distinct points in $S$, and let $m_i$ be the multiplicity of $d_i$, i.e. the number of unordered pairs $\{P,Q\}\subseteq S$ with $|PQ|=d_i$. Prove that $\sum_{i=1}^k m_i^2\leq n^3-n^2$.
2006 Baltic Way, 10
$162$ pluses and $144$ minuses are placed in a $30\times 30$ table in such a way that each row and each column contains at most $17$ signs. (No cell contains more than one sign.) For every plus we count the number of minuses in its row and for every minus we count the number of pluses in its column. Find the maximum of the sum of these numbers.