Found problems: 1800
2018 Turkey EGMO TST, 4
There are $n$ stone piles each consisting of $2018$ stones. The weight of each stone is equal to one of the numbers $1, 2, 3, ...25$ and the total weights of any two piles are different. It is given that if we choose any two piles and remove the heaviest and lightest stones from each of these piles then the pile which has the heavier one becomes the lighter one. Determine the maximal possible value of $n$.
2002 Iran Team Selection Test, 11
A $10\times10\times10$ cube has $1000$ unit cubes. $500$ of them are coloured black and $500$ of them are coloured white. Show that there are at least $100$ unit squares, being the common face of a white and a black unit cube.
1974 IMO Longlists, 34
Consider infinite diagrams
[asy]
import graph; size(90); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black;
label("$n_{00} \ n_{01} \ n_{02} \ldots$", (1.14,1.38), SE*lsf); label("$n_{10} \ n_{11} \ n_{12} \ldots$", (1.2,1.8), SE*lsf); label("$n_{20} \ n_{21} \ n_{22} \ldots$", (1.2,2.2), SE*lsf); label("$\vdots \quad \vdots \qquad \vdots $", (1.32,2.72), SE*lsf);
draw((1,1)--(3,1)); draw((1,1)--(1.02,2.62)); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle);
[/asy]
where all but a finite number of the integers $n_{ij} , i = 0, 1, 2, \ldots, j = 0, 1, 2, \ldots ,$ are equal to $0$. Three elements of a diagram are called [i]adjacent[/i] if there are integers $i$ and $j$ with $i \geq 0$ and $j \geq 0$ such that the three elements are
[b](i)[/b] $n_{ij}, n_{i,j+1}, n_{i,j+2},$ or
[b](ii)[/b] $n_{ij}, n_{i+1,j}, n_{i+2,j} ,$ or
[b](iii)[/b] $n_{i+2,j}, n_{i+1,j+1}, n_{i,j+2}.$
An elementary operation on a diagram is an operation by which three [i]adjacent[/i] elements $n_{ij}$ are changed into $n_{ij}'$ in such a way that $|n_{ij}-n_{ij}'|=1.$ Two diagrams are called equivalent if one of them can be changed into the other by a finite sequence of elementary operations. How many inequivalent diagrams exist?
2008 China Team Selection Test, 3
Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.
2001 JBMO ShortLists, 13
At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two.
[color=#BF0000]Rewording of the last line for clarification:[/color]
Find the smallest value of $k$ such that there (always) exists $3$ mathematicians $X,Y,Z$ such that $X$ and $Y$ know each other, $X$ and $Z$ know each other and $Y$ and $Z$ know each other.
2009 Indonesia TST, 1
Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.
2013 Brazil National Olympiad, 5
Let $x$ be an irrational number between 0 and 1 and $x = 0.a_1a_2a_3\cdots$ its decimal representation. For each $k \ge 1$, let $p(k)$ denote the number of distinct sequences $a_{j+1} a_{j+2} \cdots a_{j+k}$ of $k$ consecutive digits in the decimal representation of $x$. Prove that $p(k) \ge k+1$ for every positive integer $k$.
2009 Moldova Team Selection Test, 4
[color=darkblue]Let $ X$ be a group of people, where any two people are friends or enemies. Each pair of friends from $ X$ doesn't have any common friends, and any two enemies have exactly two common friends. Prove that each person from $ X$ has the same number of friends as others.[/color]
1974 IMO Longlists, 26
Let $g(k)$ be the number of partitions of a $k$-element set $M$, i.e., the number of families $\{ A_1,A_2,\ldots ,A_s\}$ of nonempty subsets of $M$ such that $A_i\cap A_j=\emptyset$ for $i\not= j$ and $\bigcup_{i=1}^n A_i=M$. Prove that, for every $n$,
\[n^n\le g(2n)\le (2n)^{2n}\]
2007 IMAR Test, 2
Denote by $ \mathcal{C}$ the family of all configurations $ C$ of $ N > 1$ distinct
points on the sphere $ S^2,$ and by $ \mathcal{H}$ the family of all closed hemispheres $ H$
of $ S^2.$ Compute:
$ \displaystyle\max_{H\in\mathcal{H}}\displaystyle\min_{C\in\mathcal{C}}|H\cap C|$, $ \displaystyle\min_{H\in\mathcal{H}}\displaystyle\max_{C\in\mathcal{C}}|H\cap C|$
$ \displaystyle\max_{C\in\mathcal{C}}\displaystyle\min_{H\in\mathcal{H}}|H\cap C|$ and $ \displaystyle\min_{C\in\mathcal{C}}\displaystyle\max_{H\in\mathcal{H}}|H\cap C|.$
2012 Vietnam National Olympiad, 1
For a group of 5 girls, denoted as $G_1,G_2,G_3,G_4,G_5$ and $12$ boys. There are $17$ chairs arranged in a row. The students have been grouped to sit in the seats such that the following conditions are simultaneously met:
(a) Each chair has a proper seat.
(b) The order, from left to right, of the girls seating is $G_1; G_2; G_3; G_4; G_5.$
(c) Between $G_1$ and $G_2$ there are at least three boys.
(d) Between $G_4$ and $G_5$ there are at least one boy and most four boys.
How many such arrangements are possible?
2003 Romania Team Selection Test, 6
At a math contest there are $2n$ students participating. Each of them submits a problem to the jury, which thereafter gives each students one of the $2n$ problems submitted. One says that the contest is [i]fair[/i] is there are $n$ participants which receive their problems from the other $n$ participants.
Prove that the number of distributions of the problems in order to obtain a fair contest is a perfect square.
2006 Bulgaria National Olympiad, 3
Consider a point $O$ in the plane. Find all sets $S$ of at least two points in the plane such that if $A\in S$ ad $A\neq O$, then the circle with diameter $OA$ is in $S$.
[i]Nikolai Nikolov, Slavomir Dinev[/i]
2007 All-Russian Olympiad Regional Round, 8.5
There are $ 11$ coins, which are indistinguishable by sight. Nevertheless, among them there are $ 10$ geniune coins ( of weight $ 20$ g each) and one counterfeit (of weight $ 21$ g). You have a two-pan scale which is blanced when the weight in the left-hand pan is twice as much as the weight in the right-hand one. Using this scale only, find the false coin by three weighings.
2008 Pan African, 2
A set of positive integers $X$ is called [i]connected[/i] if $|X|\ge 2$ and there exist two distinct elements $m$ and $n$ of $X$ such that $m$ is a divisor of $n$.
Determine the number of connected subsets of the set $\{1,2,\ldots,10\}$.
2009 Indonesia TST, 1
Given an $ n\times n$ chessboard.
a) Find the number of rectangles on the chessboard.
b) Assume there exists an $ r\times r$ square (label $ B$) with $ r<n$ which is located on the upper left corner of the board. Define "inner border" of $ A$ as the border of $ A$ which is not the border of the chessboard. How many rectangles in $ B$ that touch exactly one inner border of $ B$?
2014 Dutch BxMO/EGMO TST, 5
Let $n$ be a positive integer. Daniel and Merlijn are playing a game. Daniel
has $k$ sheets of paper lying next to each other on a table, where $k$ is a
positive integer. On each of the sheets, he writes some of the numbers
from $1$ up to $n$ (he is allowed to write no number at all, or all numbers).
On the back of each of the sheets, he writes down the remaining numbers.
Once Daniel is finished, Merlijn can flip some of the sheets of paper (he is
allowed to flip no sheet at all, or all sheets). If Merlijn succeeds in making
all of the numbers from $1$ up to n visible at least once, then he wins.
Determine the smallest $k$ for which Merlijn can always win, regardless of
Daniel’s actions.
2011 Romania Team Selection Test, 4
Given an integer $n\ge 2$, compute $\sum_{\sigma} \textrm{sgn}(\sigma) n^{\ell(\sigma)}$, where all $n$-element permutations are considered, and where $\ell(\sigma)$ is the number of disjoint cycles in the standard decomposition of $\sigma$.
1997 Iran MO (3rd Round), 6
Let $\mathcal P$ be the set of all points in $\mathbb R^n$ with rational coordinates. For the points $A,B \in \mathcal l{P}$, one can move from $A$ to $B$ if the distance $AB$ is $1$. Prove that every point in $\mathcal l{ P}$ can be reached from any other point in $\mathcal{P}$ by a finite sequence of moves if and only if $n \geq 5$.
1997 All-Russian Olympiad, 4
In an $m\times n$ rectangular grid, where m and n are odd integers, $1\times 2$ dominoes are initially placed so as to exactly cover all but one of the $1\times 1$ squares at one corner of the grid.
It is permitted to slide a domino towards the empty square, thus exposing another square.
Show that by a sequence of such moves, we can move the empty square to any corner of the rectangle.
[i]A. Shapovalov[/i]
2006 ITAMO, 6
Alberto and Barbara play the following game. Initially, there are some piles of coins on a table. Each player in turn, starting with Albert, performs one of the two following ways:
1) take a coin from an arbitrary pile;
2) select a pile and divide it into two non-empty piles.
The winner is the player who removes the last coin on the table. Determine which player has a winning strategy with respect to the initial state.
2012 China Team Selection Test, 3
Let $a_1<a_2$ be two given integers. For any integer $n\ge 3$, let $a_n$ be the smallest integer which is larger than $a_{n-1}$ and can be uniquely represented as $a_i+a_j$, where $1\le i<j\le n-1$. Given that there are only a finite number of even numbers in $\{a_n\}$, prove that the sequence $\{a_{n+1}-a_{n}\}$ is eventually periodic, i.e. that there exist positive integers $T,N$ such that for all integers $n>N$, we have
\[a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.\]
2013 National Olympiad First Round, 4
The numbers $1,2,\dots, 49$ are written on unit squares of a $7\times 7$ chessboard such that consequtive numbers are on unit squares sharing a common edge. At most how many prime numbers can a row have?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 3
$
2005 Georgia Team Selection Test, 12
$ 30$ students participated in the mathematical Olympiad. Each of them was given $ 8$ problems to solve. Jury estimated their work with the following rule:
1) Each problem was worth $ k$ points, if it wasn't solved by exactly $ k$ students;
2) Each student received the maximum possible points in each problem or got $ 0$ in it;
Lasha got the least number of points. What's the maximal number of points he could have?
Remark: 1) means that if the problem was solved by exactly $ k$ students, than each of them got $ 30 \minus{} k$ points in it.
2002 JBMO ShortLists, 2
Positive real numbers are arranged in the form:
$ 1 \ \ \ 3 \ \ \ 6 \ \ \ 10 \ \ \ 15 ...$
$ 2 \ \ \ 5 \ \ \ 9 \ \ \ 14 ...$
$ 4 \ \ \ 8 \ \ \ 13 ...$
$ 7 \ \ \ 12 ...$
$ 11 ...$
Find the number of the line and column where the number 2002 stays.