Found problems: 1488
2009 Irish Math Olympiad, 4
At a strange party, each person knew exactly $22$ others.
For any pair of people $X$ and $Y$ who knew each other, there was no other person at the party that they both knew.
For any pair of people $X$ and $Y$ who did not know one another, there were exactly $6$ other people that they both knew.
How many people were at the party?
2003 All-Russian Olympiad, 4
A finite set of points $X$ and an equilateral triangle $T$ are given on a plane. Suppose that every subset $X'$ of $X$ with no more than $9$ elements can be covered by two images of $T$ under translations. Prove that the whole set $X$ can be covered by two images of $T$ under translations.
2012 Balkan MO Shortlist, C1
Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$
Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$
1997 Irish Math Olympiad, 3
Let $ A$ be a subset of $ \{ 0,1,2,...,1997 \}$ containing more than $ 1000$ elements. Prove that either $ A$ contains a power of $ 2$ (that is, a number of the form $ 2^k$ with $ k\equal{}0,1,2,...)$ or there exist two distinct elements $ a,b \in A$ such that $ a\plus{}b$ is a power of $ 2$.
2007 Nordic, 3
The number $10^{2007}$ is written on the blackboard. Anne and Berit play a two player game in which the player in turn performs one of the following operations:
1) replace a number $x$ on the blackboard with two integers $a,b>1$ such that $ab=x$.
2) strike off one or both of two equal numbers on the blackboard.
The person who cannot perform any operation loses.
Who has the winning strategy if Anne starts?
2000 Cono Sur Olympiad, 2
The numbers $1,2,\ldots,64$ are written in the squares of an $8\times 8$ chessboard, one number to each square. Then $2\times 2$ tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than $100$. Find, with proof, the maximum number of tiles that can be placed on the chessboard, and give an example of a distribution of the numbers $1,2,\ldots,64$ into the squares of the chessboard that admits this maximum number of tiles.
1985 IMO Longlists, 15
[i]Superchess[/i] is played on on a $12 \times 12$ board, and it uses [i]superknights[/i], which move between opposite corner cells of any $3\times4$ subboard. Is it possible for a [i]superknight[/i] to visit every other cell of a superchessboard exactly once and return to its starting cell ?
2003 Rioplatense Mathematical Olympiad, Level 3, 3
An $8\times 8$ chessboard is to be tiled (i.e., completely covered without overlapping) with pieces of the following shapes:
[asy]
unitsize(.6cm);
draw(unitsquare,linewidth(1));
draw(shift(1,0)*unitsquare,linewidth(1));
draw(shift(2,0)*unitsquare,linewidth(1));
label("\footnotesize $1\times 3$ rectangle",(1.5,0),S);
draw(shift(8,1)*unitsquare,linewidth(1));
draw(shift(9,1)*unitsquare,linewidth(1));
draw(shift(10,1)*unitsquare,linewidth(1));
draw(shift(9,0)*unitsquare,linewidth(1));
label("\footnotesize T-shaped tetromino",(9.5,0),S);
[/asy] The $1\times 3$ rectangle covers exactly three squares of the chessboard, and the T-shaped tetromino covers exactly four squares of the chessboard. [list](a) What is the maximum number of pieces that can be used?
(b) How many ways are there to tile the chessboard using this maximum number of pieces?[/list]
2002 Moldova Team Selection Test, 2
Prove that there exists a partition of the set $A = \{1^3, 2^3, \ldots , 2000^3\}$ into $19$ nonempty subsets such that the sum of elements of each subset is divisible by $2001^2$.
2008 Brazil National Olympiad, 3
The venusian prophet Zabruberson sent to his pupils a $ 10000$-letter word, each letter being $ A$ or $ E$: the [i]Zabrubic word[/i]. Their pupils consider then that for $ 1 \leq k \leq 10000$, each word comprised of $ k$ consecutive letters of the Zabrubic word is a [i]prophetic word[/i] of length $ k$. It is known that there are at most $ 7$ prophetic words of lenght $ 3$. Find the maximum number of prophetic words of length $ 10$.
2013 ELMO Shortlist, 4
Let $n$ be a positive integer. The numbers $\{1, 2, ..., n^2\}$ are placed in an $n \times n$ grid, each exactly once. The grid is said to be [i]Muirhead-able[/i] if the sum of the entries in each column is the same, but for every $1 \le i,k \le n-1$, the sum of the first $k$ entries in column $i$ is at least the sum of the first $k$ entries in column $i+1$. For which $n$ can one construct a Muirhead-able array such that the entries in each column are decreasing?
[i]Proposed by Evan Chen[/i]
2011 Pre-Preparation Course Examination, 7
prove or disprove: in a connected graph $G$, every three longest paths have a vertex in common.
1987 China Team Selection Test, 3
Let $ G$ be a simple graph with $ 2 \cdot n$ vertices and $ n^{2}+1$ edges. Show that this graph $ G$ contains a $ K_{4}-\text{one edge}$, that is, two triangles with a common edge.
2002 All-Russian Olympiad, 3
On a plane are given finitely many red and blue lines, no two parallel, such that any intersection point of two lines of the same color also lies on another line of the other color. Prove that all the lines pass through a single point.
1999 Romania Team Selection Test, 16
Let $X$ be a set with $n$ elements, and let $A_{1}$, $A_{2}$, ..., $A_{m}$ be subsets of $X$ such that:
1) $|A_{i}|=3$ for every $i\in\left\{1,2,...,m\right\}$;
2) $|A_{i}\cap A_{j}|\leq 1$ for all $i,j\in\left\{1,2,...,m\right\}$ such that $i \neq j$.
Prove that there exists a subset $A$ of $X$ such that $A$ has at least $\left[\sqrt{2n}\right]$ elements, and for every $i\in\left\{1,2,...,m\right\}$, the set $A$ does not contain $A_{i}$.
[i]Alternative formulation.[/i] Let $X$ be a finite set with $n$ elements and $A_{1},A_{2},\ldots, A_{m}$ be three-elements subsets of $X$, such that $|A_{i}\cap A_{j}|\leq 1$, for every $i\neq j$. Prove that there exists $A\subseteq X$ with $|A|\geq \lfloor \sqrt{2n}\rfloor$, such that none of $A_{i}$'s is a subset of $A$.
2019 Turkey MO (2nd round), 3
There are 2019 students in a school, and some of these students are members of different student clubs. Each student club has an advisory board consisting of 12 students who are members of that particular club. An {\em advisory meeting} (for a particular club) can be realized only when each participant is a member of that club, and moreover, each of the 12 students forming the advisory board are present among the participants. It is known that each subset of at least 12 students in this school can realize an advisory meeting for exactly one student club. Determine all possible numbers of different student clubs with exactly 27 members.
2012 South africa National Olympiad, 3
Sixty points, of which thirty are coloured red, twenty are coloured blue and ten are coloured green, are marked on a circle. These points divide the circle into sixty arcs. Each of these arcs is assigned a number according to the colours of its endpoints: an arc between a red and a green point is assigned a number $1$, an arc between a red and a blue point is assigned a number $2$, and an arc between a blue and a green point is assigned a number $3$. The arcs between two points of the same colour are assigned a number $0$. What is the greatest possible sum of all the numbers assigned to the arcs?
2010 CHKMO, 2
There are $ n$ points on the plane, no three of which are collinear. Each pair of points is joined by a red, yellow or green line. For any three points, the sides of the triangle they form consist of exactly two colours. Show that $ n<13$.
2010 Tournament Of Towns, 4
A rectangle is divided into $2\times 1$ and $1\times 2$ dominoes. In each domino, a diagonal is drawn, and no two diagonals have common endpoints. Prove that exactly two corners of the rectangle are endpoints of these diagonals.
2014 European Mathematical Cup, 2
In each vertex of a regular $n$-gon $A_1A_2...A_n$ there is a unique pawn. In each step it is allowed:
1. to move all pawns one step in the clockwise direction or
2. to swap the pawns at vertices $A_1$ and $A_2$.
Prove that by a fi nite series of such steps it is possible to swap the pawns at vertices:
a) $A_i$ and $A_{i+1}$ for any $ 1 \leq i < n$ while leaving all other pawns in their initial place
b) $A_i$ and $A_j$ for any $ 1 \leq i < j \leq n$ leaving all other pawns in their initial place.
[i]Proposed by Matija Bucic[/i]
2020 Latvia TST, 1.5
Given a $6\times 6$ square consisting of unit squares, denote its rows and columns from $1$ to $6$. Figure [i]p-horse[/i] can move from square $(x; y)$ to $(x’; y’)$ if and only if both $x + x’$ and $y + y’$ are primes. At the start the [i]p-horse[/i] is located in one of the unit squares.
$a)$ Can the [i]p-horse[/i] visit every unit square exactly once?
$b$) Can the [i]p-horse[/i] visit every unit square exactly once and with the last move return to the initial starting position?
2002 All-Russian Olympiad, 3
On a plane are given $6$ red, $6$ blue, and $6$ green points, such that no three of the given points lie on a line. Prove that the sum of the areas of the triangles whose vertices are of the same color does not exceed quarter the sum of the areas of all triangles with vertices in the given points.
2008 Bundeswettbewerb Mathematik, 1
Fedja used matches to put down the equally long sides of a parallelogram whose vertices are not on a common line. He figures out that exactly 7 or 9 matches, respectively, fit into the diagonals. How many matches compose the parallelogram's perimeter?
2011 BMO TST, 5
The sweeties shop called "Olympiad" sells boxes of $6,9$ or $20$ chocolates. Groups of students from a school that is near the shop collect money to buy a chocolate for each student; to make this they buy a box and than give to everybody a chocolate. Like this students can create groups of $15=6+9$ students, $38=2*9+20$ students, etc. The seller has promised to the students that he can satisfy any group of students, and if he will need to open a new box of chocolate for any group (like groups of $4,7$ or $10$ students) than he will give all the chocolates for free to this group. Can there be constructed the biggest group that profits free chocolates, and if so, how many students are there in this group?
2010 Brazil National Olympiad, 2
Determine all values of $n$ for which there is a set $S$ with $n$ points, with no 3 collinear, with the following property: it is possible to paint all points of $S$ in such a way that all angles determined by three points in $S$, all of the same color or of three different colors, aren't obtuse. The number of colors available is unlimited.