Found problems: 1800
2001 Romania Team Selection Test, 4
Three schools have $200$ students each. Every student has at least one friend in each school (if the student $a$ is a friend of the student $b$ then $b$ is a friend of $a$).
It is known that there exists a set $E$ of $300$ students (among the $600$) such that for any school $S$ and any two students $x,y\in E$ but not in $S$, the number of friends in $S$ of $x$ and $y$ are different.
Show that one can find a student in each school such that they are friends with each other.
2008 Iran MO (3rd Round), 4
Let $ S$ be a sequence that:
\[ \left\{
\begin{array}{cc}
S_0\equal{}0\hfill\\
S_1\equal{}1\hfill\\
S_n\equal{}S_{n\minus{}1}\plus{}S_{n\minus{}2}\plus{}F_n& (n>1)
\end{array}
\right.\]
such that $ F_n$ is Fibonacci sequence such that $ F_1\equal{}F_2\equal{}1$. Find $ S_n$ in terms of Fibonacci numbers.
2007 IberoAmerican, 4
In a $ 19\times 19$ board, a piece called [i]dragon[/i] moves as follows: It travels by four squares (either horizontally or vertically) and then it moves one square more in a direction perpendicular to its previous direction. It is known that, moving so, a dragon can reach every square of the board.
The [i]draconian distance[/i] between two squares is defined as the least number of moves a dragon needs to move from one square to the other.
Let $ C$ be a corner square, and $ V$ the square neighbor of $ C$ that has only a point in common with $ C$. Show that there exists a square $ X$ of the board, such that the draconian distance between $ C$ and $ X$ is greater than the draconian distance between $ C$ and $ V$.
2012 ELMO Shortlist, 4
A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles.
[i]Calvin Deng.[/i]
2007 Baltic Way, 7
A [i]squiggle[/i] is composed of six equilateral triangles with side length $1$ as shown in the figure below. Determine all possible integers $n$ such that an equilateral triangle with side length $n$ can be fully covered with [i]squiggle[/i]s (rotations and reflections of [i]squiggle[/i]s are allowed, overlappings are not).
[asy]
import graph; size(100); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black;
draw((0,0)--(0.5,1),linewidth(2pt)); draw((0.5,1)--(1,0),linewidth(2pt)); draw((0,0)--(3,0),linewidth(2pt)); draw((1.5,1)--(2,0),linewidth(2pt)); draw((2,0)--(2.5,1),linewidth(2pt)); draw((0.5,1)--(2.5,1),linewidth(2pt)); draw((1,0)--(2,2),linewidth(2pt)); draw((2,2)--(3,0),linewidth(2pt));
dot((0,0),ds); dot((1,0),ds); dot((0.5,1),ds); dot((2,0),ds); dot((1.5,1),ds); dot((3,0),ds); dot((2.5,1),ds); dot((2,2),ds); clip((-4.28,-10.96)--(-4.28,6.28)--(16.2,6.28)--(16.2,-10.96)--cycle);[/asy]
1999 Iran MO (2nd round), 3
Let $A_1,A_2,\cdots,A_n$ be $n$ distinct points on the plane ($n>1$). We consider all the segments $A_iA_j$ where $i<j\leq{n}$ and color the midpoints of them. What's the minimum number of colored points? (In fact, if $k$ colored points coincide, we count them $1$.)
2002 Bundeswettbewerb Mathematik, 3
Given a convex polyhedron with an even number of edges.
Prove that we can attach an arrow to each edge, such that for every vertex of the polyhedron, the number of the arrows ending in this vertex is even.
1998 Tournament Of Towns, 1
A $ 20\times20\times20$ block is cut up into 8000 non-overlapping unit cubes and a number is assigned to each. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $ 1\times20\times20$ slices parallel to the faces of the block contain this unit cube. Find the sume of all numbers of the cubes outside these slices.
2002 All-Russian Olympiad Regional Round, 8.2
each cells in a $9\times 9 $ grid is painted either blue or red.two cells are called [i]diagonal neighbors[/i] if their intersection is exactly a point.show that some cell has exactly two red neighbors,or exactly two blue neighbors, or both.
2009 Argentina Iberoamerican TST, 2
There are $ m \plus{} 1$ horizontal lines and $ m$ vertical lines on the plane so that $ m(m \plus{} 1)$ intersections are made.
A mark is placed at one of the $ m$ points of the lowest horizontal line.
2 players play the game of the following rules on this lines and points.
1. Each player moves a mark from a point to a point along the lines in turns.
2. The segment is erased after a mark moved along it.
3. When a player cannot make a move, then he loses.
Prove that the lead always wins the game.
PS I haven't found a student who solved it. There can be no one.
1998 Federal Competition For Advanced Students, Part 2, 1
Let $M$ be the set of the vertices of a regular hexagon, our Olympiad symbol. How many chains $\emptyset \subset A \subset B \subset C \subset D \subset M$ of six different set, beginning with the empty set and ending with the $M$, are there?
1986 IMO Longlists, 66
One hundred red points and one hundred blue points are chosen in the plane, no three of them lying on a line. Show that these points can be connected pairwise, red ones with blue ones, by disjoint line segments.
2008 Tournament Of Towns, 4
Baron Munchausen claims that he got a map of a country that consists of five cities. Each two cities are connected by a direct road. Each road intersects no more than one another road (and no more than once). On the map, the roads are colored in yellow or red, and while circling any city (along its border) one can notice that the colors of crossed roads alternate. Can Baron's claim be true?
2000 Iran MO (2nd round), 3
Let $M=\{1,2,3,\ldots, 10000\}.$ Prove that there are $16$ subsets of $M$ such that for every $a \in M,$ there exist $8$ of those subsets that intersection of the sets is exactly $\{a\}.$
2011 Singapore Senior Math Olympiad, 4
Let $n$ and $k$ be positive integers with $n\geq k\geq 2$. For $i=1,\dots,n$, let $S_i$ be a nonempty set of consecutive integers such that among any $k$ of them, there are two with nonempty intersection. Prove that there is a set $X$ of $k-1$ integers such that each $S_i$, $i=1,\dots,n$ contains at least one integer in $X$.
2013 Federal Competition For Advanced Students, Part 1, 3
Arrange the positive integers into two lines as follows:
\begin{align*} 1 \quad 3 \qquad 6 \qquad\qquad\quad 11 \qquad\qquad\qquad\qquad\quad\ 19\qquad\qquad32\qquad\qquad 53\ldots\\
\mbox{\ \ } 2 \quad 4\ \ 5 \quad 7\ \ 8\ \ 9\ \ 10\quad\ 12\ 13\ 14\ 15\ 16\ 17\ 18\quad\ 20 \mbox{ to } 31\quad\ 33 \mbox{ to } 52\quad\ \ldots\end{align*} We start with writing $1$ in the upper line, $2$ in the lower line and $3$ again in the upper line. Afterwards, we alternately write one single integer in the upper line and a block of integers in the lower line. The number of consecutive integers in a block is determined by the first number in the previous block.
Let $a_1$, $a_2$, $a_3$, $\ldots$ be the numbers in the upper line. Give an explicit formula for $a_n$.
2007 Hong kong National Olympiad, 3
There are $2007$ boys and $2007$ girls in a middle school,every student can attend no more than $100$ academic meetings,if we know any pair of students with different sex attend at least one common meeting.prove that there must be a meeting with at least $11$ boys and $11$ girls attend.
2006 Junior Balkan Team Selection Tests - Moldova, 4
Let $n$ be a positive integer, $n\geq 4$. $n$ cards are arranged on a circle and the numbers $1$ or $-1$ are written on each of the cards. in a $question$ we may find out the product of the numbers on any $3$ cards. What is the minimum numbers if questions needed to find out the product of all $n$ numbers?
2008 Kazakhstan National Olympiad, 1
Let $ F_n$ be a set of all possible connected figures, that consist of $ n$ unit cells. For each element $ f_n$ of this set, let $ S(f_n)$ be the area of that minimal rectangle that covers $ f_n$ and each side of the rectangle is parallel to the corresponding side of the cell. Find $ max(S(f_n))$,where $ f_n\in F_n$?
Remark: Two cells are called connected if they have a common edge.
2013 National Olympiad First Round, 24
$77$ stones weighing $1,2,\dots, 77$ grams are divided into $k$ groups such that total weights of each group are different from each other and each group contains less stones than groups with smaller total weights. For how many $k\in \{9,10,11,12\}$, is such a division possible?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ \text{None of above}
$
2014 Contests, 3
A real number $f(X)\neq 0$ is assigned to each point $X$ in the space.
It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have :
\[ f(O)=f(A)f(B)f(C)f(D). \]
Prove that $f(X)=1$ for all points $X$.
[i]Proposed by Aleksandar Ivanov[/i]
2010 Iran MO (3rd Round), 2
suppose that $\mathcal F\subseteq \bigcup_{j=k+1}^{n}X^{(j)}$ and $|X|=n$. we know that $\mathcal F$ is a sperner family and it's also $H_k$. prove that:
$\sum_{B\in \mathcal F}\frac{1}{\dbinom{n-1}{|B|-1}}\le 1$
(15 points)
2009 Iran MO (2nd Round), 2
In some of the $ 1\times1 $ squares of a square garden $ 50\times50 $ we've grown apple, pomegranate and peach trees (At most one tree in each square). We call a $ 1\times1 $ square a [i]room[/i] and call two rooms [i]neighbor[/i] if they have one common side. We know that a pomegranate tree has at least one apple neighbor room and a peach tree has at least one apple neighbor room and one pomegranate neighbor room. We also know that an empty room (a room in which there’s no trees) has at least one apple neighbor room and one pomegranate neighbor room and one peach neighbor room.
Prove that the number of empty rooms is not greater than $ 1000. $
2000 Taiwan National Olympiad, 3
Consider the set $S=\{ 1,2,\ldots ,100\}$ and the family $\mathcal{P}=\{ T\subset S\mid |T|=49\}$. Each $T\in\mathcal{P}$ is labelled by an arbitrary number from $S$. Prove that there exists a subset $M$ of $S$ with $|M|=50$ such that for each $x\in M$, the set $M\backslash\{ x\}$ is not labelled by $x$.
2019 Stars of Mathematics, 3
On a board the numbers $(n-1, n, n+1)$ are written where $n$ is positive integer. On a move choose 2 numbers $a$ and $b$, delete them and write $2a-b$ and $2b-a$. After a succession of moves, on the board there are 2 zeros. Find all possible values for $n$.
Proposed by Andrei Eckstein