Found problems: 1488
2000 All-Russian Olympiad, 8
All points in a $100 \times 100$ array are colored in one of four colors red, green, blue or yellow in such a way that there are $25$ points of each color in each row and in any column. Prove that there are two rows and two columns such that their four intersection points are all in different colors.
1994 India National Olympiad, 4
Find the number of nondegenerate triangles whose vertices lie in the set of points $(s,t)$ in the plane such that $0 \leq s \leq 4$, $0 \leq t \leq 4$, $s$ and $t$ are integers.
2018 Latvia Baltic Way TST, P8
Let natural $n \ge 2$ be given. Let Laura be a student in a class of more than $n+2$ students, all of which participated in an olympiad and solved some problems. Additionally, it is known that:
[list]
[*] for every pair of students there is exactly one problem that was solved by both students;
[*] for every pair of problems there is exactly one student who solved both of them;
[*] one specific problem was solved by Laura and exactly $n$ other students.
[/list]
Determine the number of students in Laura's class.
1998 Austrian-Polish Competition, 2
For n points \[ P_1;P_2;...;P_n \] in that order on a straight line. We colored each point by 1 in 5 white, red, green, blue, and purple. A coloring is called acceptable if two consecutive points \[ P_i;P_{i+1} (i=1;2;...n-1) \] is the same color or 2 points with at least one of 2 points are colored white. How many ways acceptable color?
2014 Contests, 3
Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$, there are only finitely many values of $j$ such that $A_j\subseteq A_i$. Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$.
2006 Kurschak Competition, 3
We deal $n-1$ cards in some way to $n$ people sitting around a table. From then on, in one move a person with at least $2$ cards gives one card to each of his/her neighbors. Prove that eventually a state will be reached where everyone has at most one card.
2008 Kurschak Competition, 2
Let $n\ge 1$ and $a_1<a_2<\dots<a_n$ be integers. Let $S$ be the set of pairs $1\le i<j\le n$ for which $a_j-a_i$ is a power of $2$, and $T$ be the set of pairs $1\le i<j\le n$ with $j-i$ a power of $2$. (Here, the powers of $2$ are $1,2,4,\dots$.) Prove that $|S|\le |T|$.
2004 Baltic Way, 15
A circle is divided into $13$ segments, numbered consecutively from $1$ to $13$. Five fleas called $A,B,C,D$ and $E$ are sitting in the segments $1,2,3,4$ and $5$. A flea is allowed to jump to an empty segment five positions away in either direction around the circle. Only one flea jumps at the same time, and two fleas cannot be in the same segment. After some jumps, the fleas are back in the segments $1,2,3,4,5$, but possibly in some other order than they started. Which orders are possible ?
1991 APMO, 4
During a break, $n$ children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule:
He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on.
Determine the values of $n$ for which eventually, perhaps after many rounds, all children will have at least one candy each.
2004 Irish Math Olympiad, 2
Each of the players in a tennis tournament played one match against each of
the others. If every player won at least one match, show that there is a group
A; B; C of three players for which A beat B, B beat C and C beat A.
2010 India National Olympiad, 6
Define a sequence $ < a_n > _{n\geq0}$ by $ a_0 \equal{} 0$, $ a_1 \equal{} 1$ and
\[ a_n \equal{} 2a_{n \minus{} 1} \plus{} a_{n \minus{} 2},\]
for $ n\geq2.$
$ (a)$ For every $ m > 0$ and $ 0\leq j\leq m,$ prove that $ 2a_m$ divides
$ a_{m \plus{} j} \plus{} ( \minus{} 1)^ja_{m \minus{} j}$.
$ (b)$ Suppose $ 2^k$ divides $ n$ for some natural numbers $ n$ and $ k$. Prove that $ 2^k$ divides $ a_n.$
2007 Germany Team Selection Test, 1
Let $ n > 1, n \in \mathbb{Z}$ and $ B \equal{}\{1,2,\ldots, 2^n\}.$ A subset $ A$ of $ B$ is called weird if it contains exactly one of the distinct elements $ x,y \in B$ such that the sum of $ x$ and $ y$ is a power of two. How many weird subsets does $ B$ have?
2019 Tournament Of Towns, 2
Consider 2n+1 coins lying in a circle. At the beginning, all the coins are heads up. Moving clockwise, 2n+1 flips are performed: one coin is flipped, the next coin is skipped, the next coin is flipped, the next two coins are skipped, the next coin is flipped,the next three coins are skipped and so on, until finally 2n coins are skipped and the next coin is flipped.Prove that at the end of this procedure,exactly one coin is heads down.
2014 India IMO Training Camp, 2
Let $n$ be a natural number.A triangulation of a convex n-gon is a division of the polygon into $n-2$ triangles by drawing $n-3$ diagonals no two of which intersect at an interior point of the polygon.Let $f(n)$ denote the number of triangulations of a regular n-gon such that each of the triangles formed is isosceles.Determine $f(n)$ in terms of $n$.
2009 Vietnam Team Selection Test, 3
There are $ 6n \plus{} 4$ mathematicians participating in a conference which includes $ 2n \plus{} 1$ meetings. Each meeting has one round table that suits for $ 4$ people and $ n$ round tables that each table suits for $ 6$ people. We have known that two arbitrary people sit next to or have opposite places doesn't exceed one time.
1. Determine whether or not there is the case $ n \equal{} 1$.
2. Determine whether or not there is the case $ n > 1$.
2001 Vietnam National Olympiad, 3
$(a_{1}, a_{2}, ... , a_{2n})$ is a permutation of $\{1, 2, ... , 2n\}$ such that $|a_{i}-a_{i+1}| \neq |a_{j}-a_{j+1}|$ for $i \neq j$. Show that $a_{1}= a_{2n}+n$ iff $1 \leq a_{2i}\leq n$ for $i = 1, 2, ... n.$
1997 China Team Selection Test, 3
There are 1997 pieces of medicine. Three bottles $A, B, C$ can contain at most 1997, 97, 19 pieces of medicine respectively. At first, all 1997 pieces are placed in bottle $A$, and the three bottles are closed. Each piece of medicine can be split into 100 part. When a bottle is opened, all pieces of medicine in that bottle lose a part each. A man wishes to consume all the medicine. However, he can only open each of the bottles at most once each day, consume one piece of medicine, move some pieces between the bottles, and close them. At least how many parts will be lost by the time he finishes consuming all the medicine?
2006 Bulgaria Team Selection Test, 1
[b]Problem 1. [/b]In the cells of square table are written the numbers $1$, $0$ or $-1$ so that in every line there is exactly one $1$, amd exactly one $-1$. Each turn we change the places of two columns or two rows. Is it possible, from any such table, after finite number of turns to obtain its opposite table (two tables are opposite if the sum of the numbers written in any two corresponding squares is zero)?
[i] Emil Kolev[/i]
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]
2010 Tournament Of Towns, 5
A circle is divided by $2N$ points into $2N$ arcs of length $1$. These points are joined in pairs to form $N$ chords. Each chord divides the circle into two arcs, the length of each being an even integer. Prove that $N$ is even.
2013 Iran MO (3rd Round), 1
An $n$-stick is a connected figure consisting of $n$ matches of length $1$ which are placed horizontally or vertically and no two touch each other at points other than their ends. Two shapes that can be transformed into each other by moving, rotating or flipping are considered the same.
An $n$-mino is a shape which is built by connecting $n$ squares of side length 1 on their sides such that there's a path on the squares between each two squares of the $n$-mino.
Let $S_n$ be the number of $n$-sticks and $M_n$ the number of $n$-minos, e.g. $S_3=5$ And $M_3=2$.
(a) Prove that for any natural $n$, $S_n \geq M_{n+1}$.
(b) Prove that for large enough $n$ we have $(2.4)^n \leq S_n \leq (16)^n$.
A [b]grid segment[/b] is a segment on the plane of length 1 which it's both ends are integer points. A polystick is called [b]wise[/b] if using it and it's rotations or flips we can cover all grid segments without overlapping, otherwise it's called [b]unwise[/b].
(c) Prove that there are at least $2^{n-6}$ different unwise $n$-sticks.
(d) Prove that any polystick which is in form of a path only going up and right is wise.
(e) Extra points: Prove that for large enough $n$ we have $3^n \leq S_n \leq 12^n$
Time allowed for this exam was 2 hours.
1999 Hong kong National Olympiad, 3
Students have taken a test paper in each of $n \ge 3$ subjects. It is known that in any subject exactly three students got the best score, and for any two subjects exactly one student got the best scores in both subjects. Find the smallest $n$ for which the above conditions imply that exactly one student got the best score in each of the $n$ subjects.
2008 Germany Team Selection Test, 2
[b](i)[/b] Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 3-clique (3 nodes joined pairwise by edges).
[b](ii)[/b] Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 4-clique (4 nodes joined pairwise by edges).
2000 France Team Selection Test, 1
Some squares of a $1999\times 1999$ board are occupied with pawns. Find the smallest number of pawns for which it is possible that for each empty square, the total number of pawns in the row or column of that square is at least $1999$.
2007 All-Russian Olympiad, 5
The distance between Maykop and Belorechensk is $24$ km. Two of three friends need to reach Belorechensk from Maykop and another friend wants to reach Maykop from Belorechensk. They have only one bike, which is initially in Maykop. Each guy may go on foot (with velocity at most $6$ kmph) or on a bike (with velocity at most $18$ kmph). It is forbidden to leave a bike on a road. Prove that all of them may achieve their goals after $2$ hours $40$ minutes. (Only one guy may seat on the bike simultaneously).
[i]Folclore[/i]