Found problems: 1800
2004 Regional Olympiad - Republic of Srpska, 4
A convex $n$-gon $A_1A_2\dots A_n$ $(n>3)$ is divided into triangles by non-intersecting diagonals.
For every vertex the number of sides issuing from it is even, except for the vertices
$A_{i_1},A_{i_2},\dots,A_{i_k}$, where $1\leq i_1<\dots<i_k\leq n$. Prove that $k$ is even and
\[n\equiv i_1-i_2+\dots+i_{k-1}-i_k\pmod3\]
if $k>0$ and
\[n\equiv0\pmod3\mbox{ for }k=0.\]
Note that this leads to generalization of one recent Tournament of towns problem about triangulating of square.
2024 Austrian MO National Competition, 3
Let $n \ge 3$ be an integer. A [i]circle dance[/i] is a dance that is performed according to the following rule: On the floor, $n$ points are marked at equal distances along a large circle.
At each of these points is a sheet of paper with an arrow pointing either clockwise or counterclockwise. One of the points is labeled "Start". The dancer starts at this point. In each step, he first changes the direction of the arrow at his current position and then moves to the next point in the new direction of the arrow.
a) Show that each circle dance visits each point infinitely often.
b) How many different circle dances are there? Two circle dances are considered to be the same if they differ only by a finite number of steps at the beginning and then always visit the same points in the same order. (The common sequence of steps may begin at different times in the two dances.)
[i](Birgit Vera Schmidt)[/i]
2011 Bulgaria National Olympiad, 3
In the interior of the convex 2011-gon are $2011$ points, such that no three among the given $4022$ points (the interior points and the vertices) are collinear. The points are coloured one of two different colours and a colouring is called "good" if some of the points can be joined in such a way that the following conditions are satisfied:
1) Each segment joins two points of the same colour.
2) None of the line segments intersect.
3) For any two points of the same colour there exists a path of segments connecting them.
Find the number of "good" colourings.
2012 Cono Sur Olympiad, 5
5. $A$ and $B$ play alternating turns on a $2012 \times 2013$ board with enough pieces of the following types:
Type $1$: Piece like Type $2$ but with one square at the right of the bottom square.
Type $2$: Piece of $2$ consecutive squares, one over another.
Type $3$: Piece of $1$ square.
At his turn, $A$ must put a piece of the type $1$ on available squares of the board. $B$, at his turn, must put exactly one piece of each type on available squares of the board. The player that cannot do more movements loses. If $A$ starts playing, decide who has a winning strategy.
Note: The pieces can be rotated but cannot overlap; they cannot be out of the board. The pieces of the types $1$, $2$ and $3$ can be put on exactly $3$, $2$ and $1$ squares of the board respectively.
2012 ELMO Shortlist, 1
Let $n\ge2$ be a positive integer. Given a sequence $\left(s_i\right)$ of $n$ distinct real numbers, define the "class" of the sequence to be the sequence $\left(a_1,a_2,\ldots,a_{n-1}\right)$, where $a_i$ is $1$ if $s_{i+1} > s_i$ and $-1$ otherwise.
Find the smallest integer $m$ such that there exists a sequence $\left(w_i\right)$ of length $m$ such that for every possible class of a sequence of length $n$, there is a subsequence of $\left(w_i\right)$ that has that class.
[i]David Yang.[/i]
2006 Pre-Preparation Course Examination, 2
If $f(x)$ is the generating function of the sequence $a_1,a_2,\ldots$ and if $f(x)=\frac{r(x)}{s(x)}$ holds such that $r(x)$ and $s(x)$ are polynomials show that $a_n$ has a homogenous recurrence.
2003 Tournament Of Towns, 5
Prove that one can cut $a \times b$ rectangle, $\frac{b}{2} < a < b$, into three pieces and rearrange them into a square (without overlaps and holes).
1986 IMO Longlists, 6
In an urn there are one ball marked $1$, two balls marked $2$, and so on, up to $n$ balls marked $n$. Two balls are randomly drawn without replacement. Find the probability that the two balls are assigned the same number.
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]
2002 Iran Team Selection Test, 5
A school has $n$ students and $k$ classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Prove that we can divide students into $n-k+1$ parts taht students in each part are not friends.
2010 Iran MO (3rd Round), 4
[b]carpeting[/b]
suppose that $S$ is a figure in the plane such that it's border doesn't contain any lattice points. suppose that $x,y$ are two lattice points with the distance $1$ (we call a point lattice point if it's coordinates are integers). suppose that we can cover the plane with copies of $S$ such that $x,y$ always go on lattice points ( you can rotate or reverse copies of $S$). prove that the area of $S$ is equal to lattice points inside it.
time allowed for this question was 1 hour.
1996 Balkan MO, 4
Suppse that $X=\{1,2, \ldots, 2^{1996}-1\}$, prove that there exist a subset $A$ that satisfies these conditions:
a) $1\in A$ and $2^{1996}-1\in A$;
b) Every element of $A$ except $1$ is equal to the sum of two (possibly equal) elements from $A$;
c) The maximum number of elements of $A$ is $2012$.
[i]Romania[/i]
2004 Tuymaada Olympiad, 2
In the plane are given 100 lines such that no 2 are parallel and no 3 meet in a point. The intersection points are marked. Then all the lines and k of the marked points are erased. Given the remained points of intersection for what max k one can reconstruct the lines?
[i]Proposed by A. Golovanov[/i]
1982 IMO Longlists, 40
We consider a game on an infinite chessboard similar to that of solitaire: If two adjacent fields are occupied by pawns and the next field is empty (the three fields lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third field. Prove that if in the initial position pawns fill a $3k \times n$ rectangle, then it is impossible to reach a position with only one pawn on the board.
2010 Contests, 4
the code system of a new 'MO lock' is a regular $n$-gon,each vertex labelled a number $0$ or $1$ and coloured red or blue.it is known that for any two adjacent vertices,either their numbers or colours coincide.
find the number of all possible codes(in terms of $n$).
2001 Baltic Way, 4
Let $p$ and $q$ be two different primes. Prove that
\[\left\lfloor\frac{p}{q}\right\rfloor+\left\lfloor\frac{2p}{q}\right\rfloor+\left\lfloor\frac{3p}{q}\right\rfloor+\ldots +\left\lfloor\frac{(q-1)p}{q}\right\rfloor=\frac{1}{2}(p-1)(q-1) \]
2010 Turkey Team Selection Test, 3
Let $\Lambda$ be the set of points in the plane whose coordinates are integers and let $F$ be the collection of all functions from $\Lambda$ to $\{1,-1\}.$ We call a function $f$ in $F$ [i]perfect[/i] if every function $g$ in $F$ that differs from $f$ at finitely many points satisfies the condition
\[ \sum_{0<d(P,Q)<2010} \frac{f(P)f(Q)-g(P)g(Q)}{d(P,Q)} \geq 0 \]
where $d(P,Q)$ denotes the distance between $P$ and $Q.$ Show that there exist infinitely many [i]perfect[/i] functions that are not translates of each other.
1996 Romania Team Selection Test, 12
Let $ n\geq 3 $ be an integer and let $ p\geq 2n-3 $ be a prime number. For a set $ M $ of $ n $ points in the plane, no 3 collinear, let $ f: M\to \{0,1,\ldots, p-1\} $ be a function such that
(i) exactly one point of $ M $ maps to 0,
(ii) if a circle $ \mathcal{C} $ passes through 3 distinct points of $ A,B,C\in M $ then $ \sum_{P\in M\cap \mathcal{C}} f(P) \equiv 0 \pmod p $.
Prove that all the points in $ M $ lie on a circle.
2007 Kyiv Mathematical Festival, 4
The vertices of 100-gon (i.e., polygon with 100 sides) are colored alternately white or black. One of the vertices contains a checker. Two players in turn do two things: move the checker into other vertice along the side of 100-gon and then erase some side. The game ends when it is impossible to move the checker. At the end of the game if the checker is in the white vertice then the first player wins. Otherwise the second player wins. Does any of the players have winning strategy? If yes, then who?
[i]Remark.[/i] The answer may depend on initial position of the checker.
2005 Germany Team Selection Test, 1
In the following, a [i]word[/i] will mean a finite sequence of letters "$a$" and "$b$". The [i]length[/i] of a word will mean the number of the letters of the word. For instance, $abaab$ is a word of length $5$. There exists exactly one word of length $0$, namely the empty word.
A word $w$ of length $\ell$ consisting of the letters $x_1$, $x_2$, ..., $x_{\ell}$ in this order is called a [i]palindrome[/i] if and only if $x_j=x_{\ell+1-j}$ holds for every $j$ such that $1\leq j\leq\ell$. For instance, $baaab$ is a palindrome; so is the empty word.
For two words $w_1$ and $w_2$, let $w_1w_2$ denote the word formed by writing the word $w_2$ directly after the word $w_1$. For instance, if $w_1=baa$ and $w_2=bb$, then $w_1w_2=baabb$.
Let $r$, $s$, $t$ be nonnegative integers satisfying $r + s = t + 2$. Prove that there exist palindromes $A$, $B$, $C$ with lengths $r$, $s$, $t$, respectively, such that $AB=Cab$, if and only if the integers $r + 2$ and $s - 2$ are coprime.
2012 Romanian Masters In Mathematics, 5
Given a positive integer $n\ge 3$, colour each cell of an $n\times n$ square array with one of $\lfloor (n+2)^2/3\rfloor$ colours, each colour being used at least once. Prove that there is some $1\times 3$ or $3\times 1$ rectangular subarray whose three cells are coloured with three different colours.
[i](Russia) Ilya Bogdanov, Grigory Chelnokov, Dmitry Khramtsov[/i]
2011 Argentina Team Selection Test, 1
Each number from the set $\{1,2,3,4,5,6,7,8\}$ is either colored red or blue, following these rules:
a) The number $4$ is colored red, and there is at least one number colored blue.
b) If two numbers $x, y$ have different colors and $x + y \leq 8$, then the number $x + y$ is colored blue.
c) If two numbers $x, y$ have different colors and $x \cdot y \leq 8$, then the number $x \cdot y$ is colored red.
Find all possible ways the numbers from this set can be colored.
1997 Pre-Preparation Course Examination, 6
We have considered an arbitrary segment from each line in a plane. Show that the set of points of these segments have a subset such that the points of this subset form a triangle in the plane.
Oliforum Contest IV 2013, 8
Two distinct real numbers are written on each vertex of a convex $2012-$gon. Show that we can remove a number from each vertex such that the remaining numbers on any two adjacent vertices are different.
2012 Federal Competition For Advanced Students, Part 1, 3
Consider a stripe of $n$ fieds, numbered from left to right with the integers $1$ to $n$ in ascending order. Each of the fields is colored with one of the colors $1$, $2$ or $3$. Even-numbered fields can be colored with any color. Odd-numbered fields are only allowed to be colored with the odd colors $1$ and $3$.
How many such colorings are there such that any two neighboring fields have different colors?