Found problems: 1800
1997 Romania Team Selection Test, 4
Let $p,q,r$ be distinct prime numbers and let
\[A=\{p^aq^br^c\mid 0\le a,b,c\le 5\} \]
Find the least $n\in\mathbb{N}$ such that for any $B\subset A$ where $|B|=n$, has elements $x$ and $y$ such that $x$ divides $y$.
[i]Ioan Tomescu[/i]
2024 Czech-Polish-Slovak Junior Match, 6
We are given a rectangular table with a positive integer written in each of its cells. For each cell of the table, the number in it is equal to the total number of different values in the cells that are in the same row or column (including itself). Find all tables with this property.
2007 Romania National Olympiad, 3
The plane is divided into strips of width $1$ by parallel lines (a strip - the region between two parallel lines). The points from the interior of each strip are coloured with red or white, such that in each strip only one color is used (the points of a strip are coloured with the same color). The points on the lines are not coloured. Show that there is an equilateral triangle of side-length $100$, with all vertices of the same colour.
1998 Brazil Team Selection Test, Problem 2
Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.
1999 China National Olympiad, 3
A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings.
2007 Vietnam Team Selection Test, 1
Given two sets $A, B$ of positive real numbers such that: $|A| = |B| =n$; $A \neq B$ and $S(A)=S(B)$, where $|X|$ is the number of elements and $S(X)$ is the sum of all elements in set $X$. Prove that we can fill in each unit square of a $n\times n$ square with positive numbers and some zeros such that:
a) the set of the sum of all numbers in each row equals $A$;
b) the set of the sum of all numbers in each column equals $A$.
c) there are at least $(n-1)^{2}+k$ zero numbers in the $n\times n$ array with $k=|A \cap B|$.
2013 China Girls Math Olympiad, 6
Let $S$ be a subset of $\{0,1,2,\ldots,98 \}$ with exactly $m\geq 3$ (distinct) elements, such that for any $x,y\in S$ there exists $z\in S$ satisfying $x+y \equiv 2z \pmod{99}$. Determine all possible values of $m$.
2000 Romania Team Selection Test, 3
Determine all pairs $(m,n)$ of positive integers such that a $m\times n$ rectangle can be tiled with L-trominoes.
2009 Kazakhstan National Olympiad, 3
In chess tournament participates $n$ participants ($n >1$). In tournament each of participants plays with each other exactly $1$ game. For each game participant have $1$ point if he wins game, $0,5$ point if game is drow and $0$ points if he lose game.
If after ending of tournament participant have at least $ 75 %
$ of maximum possible points he called $winner$ $of$ $tournament$.
Find maximum possible numbers of $winners$ $of$ $tournament$.
1998 Iran MO (2nd round), 3
Let $n$ be a positive integer. We call $(a_1,a_2,\cdots,a_n)$ a [i]good[/i] $n-$tuple if $\sum_{i=1}^{n}{a_i}=2n$ and there doesn't exist a set of $a_i$s such that the sum of them is equal to $n$. Find all [i]good[/i] $n-$tuple.
(For instance, $(1,1,4)$ is a [i]good[/i] $3-$tuple, but $(1,2,1,2,4)$ is not a [i]good[/i] $5-$tuple.)
2006 Taiwan TST Round 1, 3
Every square on a $n\times n$ chessboard is colored with red, blue, or green. Each red square has at least one green square adjacent to it, each green square has at least one blue square adjacent to it, and each blue square has at least one red square adjacent to it. Let $R$ be the number of red squares. Prove that $\displaystyle \frac{n^2}{11} \le R \le \frac{2n^2}{3}$.
2009 Indonesia TST, 3
Let $ S\equal{}\{1,2,\ldots,n\}$. Let $ A$ be a subset of $ S$ such that for $ x,y\in A$, we have $ x\plus{}y\in A$ or $ x\plus{}y\minus{}n\in A$. Show that the number of elements of $ A$ divides $ n$.
2013 Vietnam National Olympiad, 4
Write down some numbers $a_1,a_2,\ldots, a_n$ from left to right on a line. Step 1, we write $a_1+a_2$ between $a_1,a_2$; $a_2+a_3$ between $a_2,a_3$, …, $a_{n-1}+a_n$ between $a_{n-1},a_n$, and then we have new sequence $b=(a_1, a_1+a_2,a_2,a_2+a_3,a_3, \ldots, a_{n-1}, a_{n-1}+a_n, a_n)$. Step 2, we do the same thing with sequence b to have the new sequence c again…. And so on. If we do 2013 steps, count the number of the number 2013 appear on the line if
a) $n=2$, $a_1=1, a_2=1000$
b) $n=1000$, $a_i=i, i=1,2\ldots, 1000$
Sorry for my bad English
[color=#008000]Moderator says: alternate phrasing here: https://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=516134[/color]
2007 Korea - Final Round, 2
Given a $ 4\times 4$ squares table. How many ways that we can fill the table with $ \{0,1\}$ such that two neighbor squares (have one common side) have product which is equal to $ 0$?
2011 Turkey Junior National Olympiad, 4
Each student chooses $1$ math problem and $1$ physics problem among $20$ math problems and $11$ physics problems. No same pair of problem is selected by two students. And at least one of the problems selected by any student is selected by at most one other student. At most how many students are there?
2008 China Team Selection Test, 4
Prove that for arbitary positive integer $ n\geq 4$, there exists a permutation of the subsets that contain at least two elements of the set $ G_{n} \equal{} \{1,2,3,\cdots,n\}$: $ P_{1},P_{2},\cdots,P_{2^n \minus{} n \minus{} 1}$ such that $ |P_{i}\cap P_{i \plus{} 1}| \equal{} 2,i \equal{} 1,2,\cdots,2^n \minus{} n \minus{} 2.$
2005 All-Russian Olympiad, 1
We select $16$ cells on an $8\times 8$ chessboard. What is the minimal number of pairs of selected cells in the same row or column?
2014 Costa Rica - Final Round, 3
There are 2014 houses in a circle. Let $A$ be one of these houses. Santa Claus enters house $A$ and leaves a gift. Then with probability $1/2$ he visits $A$'s left neighbor and with probability $1/2$ he visits $A$'s right neighbor. He leaves a gift also in that second house, and then repeats the procedure (visits with probability $1/2$ either of the neighbors, leaves a gift, etc). Santa finishes as soon as every house has received at least one gift.
Prove that any house $B$ different from $A$ has a probability of $1/2013$ of being the last house receiving a gift.
2008 Vietnam Team Selection Test, 3
Consider the set $ M = \{1,2, \ldots ,2008\}$. Paint every number in the set $ M$ with one of the three colors blue, yellow, red such that each color is utilized to paint at least one number. Define two sets:
$ S_1=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have the same color and }2008 | (x + y + z)\}$;
$ S_2=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have three pairwisely different colors and }2008 | (x + y + z)\}$.
Prove that $ 2|S_1| > |S_2|$ (where $ |X|$ denotes the number of elements in a set $ X$).
1998 Romania Team Selection Test, 1
A word of length $n$ is an ordered sequence $x_1x_2\ldots x_n$ where $x_i$ is a letter from the set $\{ a,b,c \}$. Denote by $A_n$ the set of words of length $n$ which do not contain any block $x_ix_{i+1}, i=1,2,\ldots ,n-1,$ of the form $aa$ or $bb$ and by $B_n$ the set of words of length $n$ in which none of the subsequences $x_ix_{i+1}x_{i+2}, i=1,2,\ldots n-2,$ contains all the letters $a,b,c$.
Prove that $|B_{n+1}|=3|A_n|$.
[i]Vasile Pop[/i]
2002 Federal Competition For Advanced Students, Part 2, 1
Consider all possible rectangles that can be drawn on a $8 \times 8$ chessboard, covering only whole cells. Calculate the sum of their areas.
What formula is obtained if “$8 \times 8$” is replaced with “$a \times b$”, where $a, b$ are positive integers?
2017 Baltic Way, 8
A chess knight has injured his leg and is limping. He alternates between a normal move and a short move where he moves to any diagonally neighbouring cell.
The limping knight moves on a $5 \times 6$ cell chessboard starting with a normal move. What is the largest number of moves he can make if he is starting from a cell of his own choice and is not allowed to visit any cell (including the initial cell) more than once?
2002 Baltic Way, 9
Two magicians show the following trick. The first magician goes out of the room. The second magician takes a deck of $100$ cards labelled by numbers $1,2,\ldots ,100$ and asks three spectators to choose in turn one card each. The second magician sees what card each spectator has taken. Then he adds one more card from the rest of the deck. Spectators shuffle these $4$ cards, call the first magician and give him these $4$ cards. The first magician looks at the $4$ cards and “guesses” what card was chosen by the first spectator, what card by the second and what card by the third. Prove that the magicians can perform this trick.
2007 Indonesia TST, 4
Let $ S$ be a finite family of squares on a plane such that every point on that plane is contained in at most $ k$ squares in $ S$. Prove that $ P$ can be divided into $ 4(k\minus{}1)\plus{}1$ sub-family such that in each sub-family, each pair of squares are disjoint.
2010 ELMO Shortlist, 4
The numbers $1, 2, \ldots, n$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number remains. Prove that this number is at least $\frac{4}{9}n^3$.
[i]Brian Hamrick.[/i]