Found problems: 1800
1986 IMO Longlists, 42
The integers $1, 2, \cdots, n^2$ are placed on the fields of an $n \times n$ chessboard $(n > 2)$ in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most $n + 1$. What is the total number of such placements?
2012 ELMO Shortlist, 3
Find all ordered pairs of positive integers $(m,n)$ for which there exists a set $C=\{c_1,\ldots,c_k\}$ ($k\ge1$) of colors and an assignment of colors to each of the $mn$ unit squares of a $m\times n$ grid such that for every color $c_i\in C$ and unit square $S$ of color $c_i$, exactly two direct (non-diagonal) neighbors of $S$ have color $c_i$.
[i]David Yang.[/i]
1977 IMO Longlists, 15
Let $n$ be an integer greater than $1$. In the Cartesian coordinate system we consider all squares with integer vertices $(x,y)$ such that $1\le x,y\le n$. Denote by $p_k\ (k=0,1,2,\ldots )$ the number of pairs of points that are vertices of exactly $k$ such squares. Prove that $\sum_k(k-1)p_k=0$.
2014 Contests, 1
Is it possible to fill a $3 \times 3$ grid with each of the numbers $1,2,\ldots,9$ once each such that the sum of any two numbers sharing a side is prime?
2007 QEDMO 5th, 4
Let $ n$ be a positive integer, and let $ \left( a_{1},\ a_{2} ,\ ...,\ a_{n}\right)$, $ \left( b_{1},\ b_{2},\ ...,\ b_{n}\right)$ and $ \left( c_{1},\ c_{2},\ ...,\ c_{n}\right)$ be three sequences of integers such that for any two distinct numbers $ i$ and $ j$ from the set $ \left\{ 1,2,...,n\right\}$, none of the seven integers
$ a_{i}\minus{}a_{j}$; $ \left( b_{i}\plus{}c_{i}\right) \minus{}\left( b_{j}\plus{}c_{j}\right)$;
$ b_{i}\minus{}b_{j}$; $ \left( c_{i}\plus{}a_{i}\right) \minus{}\left( c_{j}\plus{}a_{j}\right)$;
$ c_{i}\minus{}c_{j}$; $ \left( a_{i}\plus{}b_{i}\right) \minus{}\left( a_{j}\plus{}b_{j}\right)$;
$ \left( a_{i}\plus{}b_{i}\plus{}c_{i}\right) \minus{}\left( a_{j}\plus{}b_{j}\plus{}c_{j}\right)$
is divisible by $ n$.
Prove that:
[b]a)[/b] The number $ n$ is odd.
[b]b)[/b] The number $ n$ is not divisible by $ 3$.
[hide="Source of the problem"][i]Source of the problem:[/i] This question is a generalization of one direction of Theorem 2.1 in: Dean Alvis, Michael Kinyon, [i]Birkhoff's Theorem for Panstochastic Matrices[/i], American Mathematical Monthly, 1/2001 (Vol. 108), pp. 28-37. The original Theorem 2.1 is obtained if you require $ b_{i}\equal{}i$ and $ c_{i}\equal{}\minus{}i$ for all $ i$, and add in a converse stating that such sequences $ \left( a_{1},\ a_{2},\ ...,\ a_{n}\right)$, $ \left( b_{1},\ b_{2},\ ...,\ b_{n}\right)$ and $ \left( c_{1} ,\ c_{2},\ ...,\ c_{n}\right)$ indeed exist if $ n$ is odd and not divisible by $ 3$.[/hide]
1998 Turkey Team Selection Test, 3
Let $A = {1, 2, 3, 4, 5}$. Find the number of functions $f$ from the nonempty subsets of $A$ to $A$, such that $f(B) \in B$ for any $B \subset A$, and $f(B \cup C)$ is either $f(B)$ or $f(C)$ for any $B$, $C \subset A$
2008 Bulgaria Team Selection Test, 3
Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.
2005 Iran Team Selection Test, 3
Suppose there are 18 lighthouses on the Persian Gulf. Each of the lighthouses lightens an angle with size 20 degrees. Prove that we can choose the directions of the lighthouses such that whole of the blue Persian (always Persian) Gulf is lightened.
2010 Macedonia National Olympiad, 3
A total of $2010$ coins are distributed in $5$ boxes. At the beginning the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that she can do the following transformation finitely many times: from a box with at least 4 coins she can transfer one coin to each of the other boxes.
What is the maximum number of coins that Martha can take away?
1998 Tournament Of Towns, 4
Twelve candidates for mayor participate in a TV talk show. At some point a candidate said: "One lie has been told." Another said: "Now two lies have been told". "Now three lies," said a third. This continued until the twelfth said: "Now twelve lies have been told". At this point the moderator ended the discussion. It turned out that at least one of the candidates correctly stated the number of lies told before he made the claim. How many lies were actually told by the candidates?
2024 Bundeswettbewerb Mathematik, 4
For positive integers $p$, $q$ and $r$ we are given $p \cdot q \cdot r$ unit cubes. We drill a hole along the space diagonal of each of these cubes and then tie them to a very thin thread of length $p \cdot q \cdot r \cdot \sqrt{3}$ like a string of pearls.
We now want to construct a cuboid of side lengths $p$, $q$ and $r$ out of the cubes, without tearing the thread.
a) For which numbers $p$, $q$ and $r$ is this possible?
b) For which numbers $p$, $q$ and $r$ is this possible in a way such that both ends of the thread coincide?
2006 Moldova Team Selection Test, 4
Let $m$ circles intersect in points $A$ and $B$. We write numbers using the following algorithm: we write $1$ in points $A$ and $B$, in every midpoint of the open arc $AB$ we write $2$, then between every two numbers written in the midpoint we write their sum and so on repeating $n$ times. Let $r(n,m)$
be the number of appearances of the number $n$ writing all of them on our $m$ circles.
a) Determine $r(n,m)$;
b) For $n=2006$, find the smallest $m$ for which $r(n,m)$ is a perfect square.
Example for half arc: $1-1$;
$1-2-1$;
$1-3-2-3-1$;
$1-4-3-5-2-5-3-4-1$;
$1-5-4-7-3-8-5-7-2-7-5-8-3-7-4-5-1$...
2024 Abelkonkurransen Finale, 3a
Determine the smallest constant $N$ so that the following may hold true:
Geostan has deployed secret agents in Combostan. All pairs of agents can communicate, either directly or through other agents. The distance between two agents is the smallest number of agents in a communication chain between the two agents.
Andreas and Edvard are among these agents, and Combostan has given Noah the task of determining the distance between Andreas and Edvard. Noah has a list of numbers, one for each agent. The number of an agent describes the longest of the two distances from the agent to Andreas and Edvard. However, Noah does not know which number corresponds to which agent, or which agents have direct contact.
Given this information, he can write down $N$ numbers and prove that the distance between Andreas and Edvard is one of these $N$ numbers. The number $N$ is independent of the agents’ communication network.
2008 Bosnia Herzegovina Team Selection Test, 3
$ 30$ persons are sitting at round table. $ 30 \minus{} N$ of them always speak true ("true speakers") while the other $ N$ of them sometimes speak true sometimes not ("lie speakers"). Question: "Who is your right neighbour - "true speaker" or "lie speaker" ?" is asked to all 30 persons and 30 answers are collected. What is maximal number $ N$ for which (with knowledge of these answers) we can always be sure (decide) about at least one person who is "true speaker".
2016 Middle European Mathematical Olympiad, 3
A $8 \times 8$ board is given, with sides directed north-south and east-west.
It is divided into $1 \times 1$ cells in the usual manner. In each cell, there is most one [i]house[/i]. A house occupies only one cell.
A house is [i] in the shade[/i] if there is a house in each of the cells in the south, east and west sides of its cell. In particular, no house placed on the south, east or west side of the board is in the shade.
Find the maximal number of houses that can be placed on the board such that no house is in the shade.
2009 USA Team Selection Test, 6
Let $ N > M > 1$ be fixed integers. There are $ N$ people playing in a chess tournament; each pair of players plays each other once, with no draws. It turns out that for each sequence of $ M \plus{} 1$ distinct players $ P_0, P_1, \ldots P_M$ such that $ P_{i \minus{} 1}$ beat $ P_i$ for each $ i \equal{} 1, \ldots, M$, player $ P_0$ also beat $ P_M$. Prove that the players can be numbered $ 1,2, \ldots, N$ in such a way that, whenever $ a \geq b \plus{} M \minus{} 1$, player $ a$ beat player $ b$.
[i]Gabriel Carroll.[/i]
1987 IMO Longlists, 9
In the set of $20$ elements $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C, D, J, K, L, U, X, Y , Z\}$ we have made a random sequence of $28$ throws. What is the probability that the sequence $CUBA \ JULY \ 1987$ appears in this order in the sequence already thrown?
2011 Argentina Team Selection Test, 6
Each square of $1\times 1$, of a $n\times n$ grid is colored using red or blue, in such way that between all the $2\times 2$ subgrids, there are all the possible colorations of a $2\times 2$ grid using red or blue, (colorations that can be obtained by using rotation or symmetry, are said to be different, so there are 16 possibilities). Find:
a) The minimum value of $n$.
b) For that value, find the least possible number of red squares.
2010 CentroAmerican, 4
Find all positive integers $N$ such that an $N\times N$ board can be tiled using tiles of size $5\times 5$ or $1\times 3$.
Note: The tiles must completely cover all the board, with no overlappings.
1983 Canada National Olympiad, 5
The geometric mean (G.M.) of $k$ positive integers $a_1$, $a_2$, $\dots$, $a_k$ is defined to be the (positive) $k$-th root of their product. For example, the G.M. of 3, 4, 18 is 6. Show that the G.M. of a set $S$ of $n$ positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of $S$.
2021 Turkey MO (2nd round), 6
In a school, there are 2021 students, each having exactly $k$ friends. There aren't three students such that all three are friends with each other. What is the maximum possible value of $k$?
2007 Junior Tuymaada Olympiad, 5
What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?
1974 IMO Longlists, 28
Let $M$ be a finite set and $P=\{ M_1,M_2,\ldots ,M_l\}$ a partition of $M$ (i.e., $\bigcup_{i=1}^k M_i, M_i\not=\emptyset, M_i\cap M_j =\emptyset$ for all $i,j\in\{1,2, \ldots ,k\} ,i\not= j)$. We define the following elementary operation on $P$:
Choose $i,j\in\{1,2,\ldots ,k\}$, such that $i=j$ and $M_i$ has a elements and $M_j$ has $b$ elements such that $a\ge b$. Then take $b$ elements from $M_i$ and place them into $M_j$, i.e., $M_j$ becomes the union of itself and a $b$-element subset of $M_i$, while the same subset is subtracted from $M_i$ (if $a=b$, $M_i$ is thus removed from the partition).
Let a finite set $M$ be given. Prove that the property “for every partition $P$ of $M$ there exists a sequence $P=P_1,P_2,\ldots ,P_r$ such that $P_{i+1}$ is obtained from $P_i$ by an elementary operation and $P_r=\{M\}$” is equivalent to “the number of elements of $M$ is a power of $2$.”
2007 Indonesia TST, 3
On each vertex of a regular $ n\minus{}$gon there was a crow. Call this as initial configuration. At a signal, they all flew by and after a while, those $ n$ crows came back to the $ n\minus{}$gon, one crow for each vertex. Call this as final configuration. Determine all $ n$ such that: there are always three crows such that the triangle they formed in the initial configuration and the triangle they formed in the final configuration are both right-angled triangle.
2011 Pre-Preparation Course Examination, 6
We call a subset $S$ of vertices of graph $G$, $2$-dominating, if and only if for every vertex $v\notin S,v\in G$, $v$ has at least two neighbors in $S$. prove that every $r$-regular $(r\ge3)$ graph has a $2$-dominating set with size at most $\frac{n(1+\ln(r))}{r}$.(15 points)
time of this exam was 3 hours