Found problems: 1800
2009 Indonesia TST, 2
Prove that there exists two different permutations $ (a_1,a_2,\dots,a_{2009})$ and $ (b_1,b_2,\dots,b_{2009})$ of $ (1,2,\dots,2009)$ such that \[ \sum_{i\equal{}1}^{2009}i^i a_i \minus{} \sum_{i\equal{}1}^{2009} i^i b_i\] is divisible by $ 2009!$.
2002 Tournament Of Towns, 3
A convex $N\text{-gon}$ is divided by diagonals into triangles so that no two diagonals intersect inside the polygon. The triangles are painted in black and white so that any two triangles are painted in black and white so that any two triangles with a common side are painted in different colors. For each $N$ find the maximal difference between the numbers of black and white triangles.
2006 Turkey Team Selection Test, 3
Each one of 2006 students makes a list with 12 schools among 2006. If we take any 6 students, there are two schools which at least one of them is included in each of 6 lists. A list which includes at least one school from all lists is a good list.
a) Prove that we can always find a good list with 12 elements, whatever the lists are;
b) Prove that students can make lists such that no shorter list is good.
2011 Baltic Way, 8
In Greifswald there are three schools called $A,B$ and $C$, each of which is attended by at least one student. Among any three students, one from $A$, one from $B$ and one from $C$, there are two knowing each other and two not knowing each other. Prove that at least one of the following holds:
[list]
[*]Some student from $A$ knows all students from $B$.
[*]Some student from $B$ knows all students from $C$.
[*] Some student from $C$ knows all students from $A$.[/list]
2004 Bulgaria Team Selection Test, 3
A table with $m$ rows and $n$ columns is given. At any move one chooses some empty cells such that any two of them lie in different rows and columns, puts a white piece in any of those cells and then puts a black piece in the cells whose rows and columns contain white pieces. The game is over if it is not possible to make a move. Find the maximum possible number of white pieces that can be put on the table.
2008 Indonesia MO, 4
Let $ A \equal{} \{1,2,\ldots,2008\}$
a) Find the number of subset of $ A$ which satisfy : the product of its elements is divisible by 7
b) Let $ N(i)$ denotes the number of subset of $ A$ which sum of its elements remains $ i$ when divided by 7. Prove that $ N(0) \minus{} N(1) \plus{} N(2) \minus{} N(3) \plus{} N(4) \minus{} N(5) \plus{} N(6)\minus{}N(7) \equal{} 0$
EDITED : thx for cosinator.. BTW, your statement and my correction give 80% hint of the solution :D
2006 International Zhautykov Olympiad, 3
Let $ m\geq n\geq 4$ be two integers. We call a $ m\times n$ board filled with 0's or 1's [i]good[/i] if
1) not all the numbers on the board are 0 or 1;
2) the sum of all the numbers in $ 3\times 3$ sub-boards is the same;
3) the sum of all the numbers in $ 4\times 4$ sub-boards is the same.
Find all $ m,n$ such that there exists a good $ m\times n$ board.
2005 ISI B.Stat Entrance Exam, 9
Suppose that to every point of the plane a colour, either red or blue, is associated.
(a) Show that if there is no equilateral triangle with all vertices of the same colour then there must exist three points $A,B$ and $C$ of the same colour such that $B$ is the midpoint of $AC$.
(b) Show that there must be an equilateral triangle with all vertices of the same colour.
2002 France Team Selection Test, 3
Let $n$ be a positive integer and let $(a_1,a_2,\ldots ,a_{2n})$ be a permutation of $1,2,\ldots ,2n$ such that the numbers $|a_{i+1}-a_i|$ are pairwise distinct for $i=1,\ldots ,2n-1$.
Prove that $\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$ if and only if $a_1-a_{2n}=n$.
2005 Bulgaria National Olympiad, 5
For positive integers $t,a,b,$a $(t,a,b)$-[i]game[/i] is a two player game defined by the following rules. Initially, the number $t$ is written on a blackboard. At his first move, the 1st player replaces $t$ with either $t-a$ or $t-b$. Then, the 2nd player subtracts either $a$ or $b$ from this number, and writes the result on the blackboard, erasing the old number. After this, the first player once again erases either $a$ or $b$ from the number written on the blackboard, and so on. The player who first reaches a negative number loses the game. Prove that there exist infinitely many values of $t$ for which the first player has a winning strategy for all pairs $(a,b)$ with $a+b=2005$.
2005 Iran MO (2nd round), 3
In one galaxy, there exist more than one million stars. Let $M$ be the set of the distances between any $2$ of them. Prove that, in every moment, $M$ has at least $79$ members. (Suppose each star as a point.)
1997 All-Russian Olympiad, 2
A class consists of 33 students. Each student is asked how many other students in the class have his first name, and how many have his last name. It turns out that each number from 0 to 10 occurs among the answers. Show that there are two students in the class with the same first and last name.
[i]A. Shapovalov[/i]
1993 All-Russian Olympiad, 3
In a tennis tournament, $n$ players want to make $2$ vs $2$ matches such that each player has each of the other players as opponents exactly once. Find all possible values of $n$.
2009 Indonesia TST, 3
In how many ways we can choose 3 non empty and non intersecting subsets from $ (1,2,\ldots,2008)$.
2016 India National Olympiad, P4
Suppose $2016$ points of the circumference of a circle are colored red and the remaining points are colored blue . Given any natural number $n\ge 3$, prove that there is a regular $n$-sided polygon all of whose vertices are blue.
2021 Baltic Way, 10
John has a string of paper where $n$ real numbers $a_i \in [0, 1]$, for all $i \in \{1, \ldots, n\}$, are written in a row.
Show that for any given $k < n$, he can cut the string of paper into non-empty $k$ pieces, between adjacent numbers, in such a way that the sum of the numbers on each piece does not differ from any other sum by more than $1$.
2005 Georgia Team Selection Test, 1
1. The transformation $ n \to 2n \minus{} 1$ or $ n \to 3n \minus{} 1$, where $ n$ is a positive integer, is called the 'change' of $ n$. Numbers $ a$ and $ b$ are called 'similar', if there exists such positive integer, that can be got by finite number of 'changes' from both $ a$ and $ b$. Find all positive integers 'similar' to $ 2005$ and less than $ 2005$.
1982 Dutch Mathematical Olympiad, 3
Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?
2006 Iran MO (3rd Round), 5
Let $E$ be a family of subsets of $\{1,2,\ldots,n\}$ with the property that for each $A\subset \{1,2,\ldots,n\}$ there exist $B\in F$ such that $\frac{n-d}2\leq |A \bigtriangleup B| \leq \frac{n+d}2$. (where $A \bigtriangleup B = (A\setminus B) \cup (B\setminus A)$ is the symmetric difference). Denote by $f(n,d)$ the minimum cardinality of such a family.
a) Prove that if $n$ is even then $f(n,0)\leq n$.
b) Prove that if $n-d$ is even then $f(n,d)\leq \lceil \frac n{d+1}\rceil$.
c) Prove that if $n$ is even then $f(n,0) = n$
2008 China Team Selection Test, 3
Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.
1998 Romania Team Selection Test, 1
Let $n\ge 2$ be an integer. Show that there exists a subset $A\in \{1,2,\ldots ,n\}$ such that:
i) The number of elements of $A$ is at most $2\lfloor\sqrt{n}\rfloor+1$
ii) $\{ |x-y| \mid x,y\in A, x\not= y\} = \{ 1,2,\ldots n-1 \}$
[i]Radu Todor[/i]
2011 ITAMO, 3
Integers between $1$ and $7$ are written on a blackboard. It is possible that not all the numbers from 1 to 7 are present, and it is also possible that one, some or all of the numbers are repeated, one or more times.
A move consists of choosing one or more numbers on the blackboard, where all distinct, delete them and write different numbers in their place, such that the written numbers together with those deleted form the whole set $\{1, 2, 3, 4, 5, 6 , 7\}$
For example, moves allowed are:
• delete a $4$ and a $5$, and write in their place the numbers $1, 2, 3, 6$ and $7$;
• deleting a $1$, a $2$, a $3$, a $4$, a $5$, a $6$ and a $7$ and write nothing in their place.
Prove that, if it is possible to find a sequence of moves, starting from the initial situation, leading to have on board a single number (written once), then this number does not depend on the sequence of moves used.
2008 Singapore Team Selection Test, 3
Fifty teams participate in a round robin competition over 50 days. Moreover, all the teams (at least two) that show up in any day must play against each other. Prove that on every pair of consecutive days, there is a team that has to play on those two days.
2009 Indonesia TST, 2
Consider the following array:
\[ 3, 5\\3, 8, 5\\3, 11, 13, 5\\3, 14, 24, 18, 5\\3, 17, 38, 42, 23, 5\\ \ldots
\] Find the 5-th number on the $ n$-th row with $ n>5$.
2011 Pre-Preparation Course Examination, 1
We have some cards that have the same look, but at the back of some of them is written $0$ and for the others $1$.(We can't see the back of a card so we can't know what's the number on it's back). we have a machine. we give it two cards and it gives us the product of the numbers on the back of the cards. if we have $m$ cards with $0$ on their back and $n$ cards with $1$ on their back, at least how many times we must use the machine to be sure that we get the number $1$? (15 points)