Found problems: 1488
1993 Baltic Way, 12
There are $13$ cities in a certain kingdom. Between some pairs of the cities a two-way direct bus, train or plane connections are established. What is the least possible number of connections to be established so that choosing any two means of transportation one can go from any city to any other without using the third kind of vehicle?
2008 South East Mathematical Olympiad, 1
Given a set $S=\{1,2,3,\ldots,3n\},(n\in N^*)$, let $T$ be a subset of $S$, such that for any $x, y, z\in T$ (not necessarily distinct) we have $x+y+z\not \in T$. Find the maximum number of elements $T$ can have.
2008 Baltic Way, 12
In a school class with $ 3n$ children, any two children make a common present to exactly one other child. Prove that for all odd $ n$ it is possible that the following holds: For any three children $ A$, $ B$ and $ C$ in the class, if $ A$ and $ B$ make a present to $ C$ then $ A$ and $ C$ make a present to $ B$.
2005 MOP Homework, 1
Consider all binary sequences (sequences consisting of 0’s and 1’s). In such a sequence the following four types of operation are allowed: (a) $010 \rightarrow 1$, (b) $1 \rightarrow 010$, (c) $110 \rightarrow 0$, and (d) $0 \rightarrow 110$. Determine if it is possible to obtain the sequence $100...0$ (with $2003$ zeroes) from the sequence $0...01$ (with $2003$ zeroes).
2012 India Regional Mathematical Olympiad, 4
Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{5,7,8\}$.
2010 JBMO Shortlist, 1
$\textbf{Problem C.1}$
There are two piles of coins, each containing $2010$ pieces. Two players $A$ and $B$ play a game taking turns ($A$ plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?
2003 Canada National Olympiad, 5
Let $S$ be a set of $n$ points in the plane such that any two points of $S$ are at least $1$ unit apart.
Prove there is a subset $T$ of $S$ with at least $\frac{n}{7}$ points such that any two points of $T$ are at least $\sqrt{3}$ units apart.
2005 China Team Selection Test, 3
We call a matrix $\textsl{binary matrix}$ if all its entries equal to $0$ or $1$. A binary matrix is $\textsl{Good}$ if it simultaneously satisfies the following two conditions:
(1) All the entries above the main diagonal (from left to right), not including the main diagonal, are equal.
(2) All the entries below the main diagonal (from left to right), not including the main diagonal, are equal.
Given positive integer $m$, prove that there exists a positive integer $M$, such that for any positive integer $n>M$ and a given $n \times n$ binary matrix $A_n$, we can select integers $1 \leq i_1 <i_2< \cdots < i_{n-m} \leq n$ and delete the $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th rows and $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th columns of $A_n$, then the resulting binary matrix $B_m$ is $\textsl{Good}$.
2001 Tournament Of Towns, 7
Several boxes are arranged in a circle. Each box may be empty or may contain one or several chips. A move consists of taking all the chips from some box and distributing them one by one into subsequent boxes clockwise starting from the next box in the clockwise direction.
(a) Suppose that on each move (except for the first one) one must take the chips from the box where the last chip was placed on the previous move. Prove that after several moves the initial distribution of the chips among the boxes will reappear.
(b) Now, suppose that in each move one can take the chips from any box. Is it true
that for every initial distribution of the chips you can get any possible distribution?
2005 China Team Selection Test, 2
Given positive integer $n (n \geq 2)$, find the largest positive integer $\lambda$ satisfying :
For $n$ bags, if every bag contains some balls whose weights are all integer powers of $2$ (the weights of balls in a bag may not be distinct), and the total weights of balls in every bag are equal, then there exists a weight among these balls such that the total number of balls with this weight is at least $\lambda$.
2010 Albania National Olympiad, 5
All members of the senate were firstly divided into $S$ senate commissions . According to the rules, no commission has less that $5$ senators and every two commissions have different number of senators. After the first session the commissions were closed and new commissions were opened. Some of the senators now are not a part of any commission. It resulted also that every two senators that were in the same commission in the first session , are not any more in the same commission.
[b](a)[/b]Prove that at least $4S+10$ senators were left outside the commissions.
[b](b)[/b]Prove that this number is achievable.
Albanian National Mathematical Olympiad 2010---12 GRADE Question 5.
2010 Contests, 2
In a blackboard there are $K$ circles in a row such that one of the numbers $1,...,K$ is assigned to each circle from the left to the right.
Change of situation of a circle is to write in it or erase the number which is assigned to it.At the beginning no number is written in its own circle.
For every positive divisor $d$ of $K$ ,$1\leq d\leq K$ we change the situation of the circles in which their assigned numbers are divisible by $d$,performing for each divisor $d$ $K$ changes of situation.
Determine the value of $K$ for which the following holds;when this procedure is applied once for all positive divisors of $K$ ,then all numbers $1,2,3,...,K$ are written in the circles they were assigned in.
2011 India IMO Training Camp, 3
Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let
\[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\]
Prove that :
$a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b].
$b)$ the number of good subsets of $T$ is [b]odd[/b].
2006 Rioplatense Mathematical Olympiad, Level 3, 2
A given finite number of lines in the plane, no two of which are parallel and no three of which are concurrent, divide the plane into finite and infinite regions. In each finite region we write $1$ or $-1$. In one operation, we can choose any triangle made of three of the lines (which may be cut by other lines in the collection) and multiply by $-1$ each of the numbers in the triangle. Determine if it is always possible to obtain $1$ in all the finite regions by successively applying this operation, regardless of the initial distribution of $1$s and $-1$s.
2008 Bulgaria National Olympiad, 3
Let $M$ be the set of the integer numbers from the range $[-n, n]$. The subset $P$ of $M$ is called a [i]base subset[/i] if every number from $M$ can be expressed as a sum of some different numbers from $P$. Find the smallest natural number $k$ such that every $k$ numbers that belongs to $M$ form a base subset.
2006 Estonia Math Open Junior Contests, 5
A $ 9 \times 9$ square is divided into unit squares. Is it possible to fill each unit square with a number $ 1, 2,..., 9$ in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?
2014 Vietnam National Olympiad, 3
Given a regular 103-sided polygon. 79 vertices are colored red and the remaining vertices are colored blue. Let $A$ be the number of pairs of adjacent red vertices and $B$ be the number of pairs of adjacent blue vertices.
a) Find all possible values of pair $(A,B).$
b) Determine the number of pairwise non-similar colorings of the polygon satisfying $B=14.$ 2 colorings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon.
1993 Hungary-Israel Binational, 4
Find the largest possible number of rooks that can be placed on a $3n \times 3n$ chessboard so that each rook is attacked by at most one rook.
2011 Kazakhstan National Olympiad, 3
In some cells of a rectangular table $m\times n (m, n> 1)$ is one checker. $Baby$ cut along the lines of the grid this table so that it is split into two equal parts, with the number of pieces on each side were the same. $Carlson$ changed the arrangement of checkers on the board (and on each side of the cage is still worth no more than one pieces). Prove that the $Baby$ may again cut the board into two equal parts containing an equal number of pieces
1990 Vietnam National Olympiad, 3
The children sitting around a circle are playing the game as follows. At first the teacher gives each child an even number of candies (bigger than $ 0$, may be equal, maybe different). A certain child gives half of his candies to his neighbor on the right. Then the child who has just received candies does the same if he has an even number of candies, otherwise he gets one candy from the teacher and then does the job; and so on. Prove that after several steps there will be a child who will be able, giving the teacher half of his candies, to make the numbers of candies of all the children equal.
2012 Portugal MO, 3
Helena and Luis are going to play a game with two bags with marbles. They play alternately and on each turn they can do one and only one of the following moves:
[list]
Take out a marble from one bag.
Take out a marble from each bag.
Take out a marble from one bag and then put it into the other bag.
[/list]
The player who leaves both bags empty wins the game.
Before starting the game, Helena counted out the marbles of each bag and said to Luis: "You may start!", while she thought "I will certainly win...". What are the possible distributions of the marbles in the bags?
1997 Finnish National High School Mathematics Competition, 4
Count the sum of the four-digit positive integers containing only odd digits in their decimal representation.
2010 Canada National Olympiad, 1
For all natural $n$, an $n$-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to $n$ squares in the $n^{th}$ row, such that all the left-most squares in each row are aligned vertically.
Let $f(n)$ denote the minimum number of square tiles requires to tile the $n$-staircase, where the side lengths of the square tiles can be any natural number. e.g. $f(2)=3$ and $f(4)=7$.
(a) Find all $n$ such that $f(n)=n$.
(b) Find all $n$ such that $f(n) = n+1$.
1995 China Team Selection Test, 1
Let $S = \lbrace A = (a_1, \ldots, a_s) \mid a_i = 0$ or $1, i = 1, \ldots, 8 \rbrace$. For any 2 elements of $S$, $A = \lbrace a_1, \ldots, a_8\rbrace$ and $B = \lbrace b_1, \ldots, b_8\rbrace$. Let $d(A,B) = \sum_{i=1}{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most how many elements can $S$ have such that the distance between any 2 sets is at least 5?
1994 IberoAmerican, 2
Let $n$ and $r$ two positive integers. It is wanted to make $r$ subsets $A_1,\ A_2,\dots,A_r$ from the set $\{0,1,\cdots,n-1\}$ such that all those subsets contain exactly $k$ elements and such that, for all integer $x$ with $0\leq{x}\leq{n-1}$ there exist $x_1\in{}A_1,\ x_2\in{}A_2 \dots,x_r\in{}A_r$ (an element of each set) with $x=x_1+x_2+\cdots+x_r$.
Find the minimum value of $k$ in terms of $n$ and $r$.