Found problems: 1488
2005 France Team Selection Test, 3
In an international meeting of $n \geq 3$ participants, 14 languages are spoken. We know that:
- Any 3 participants speak a common language.
- No language is spoken more that by the half of the participants.
What is the least value of $n$?
2007 Germany Team Selection Test, 2
Let $ n, k \in \mathbb{N}$ with $ 1 \leq k \leq \frac {n}{2} - 1.$ There are $ n$ points given on a circle. Arbitrarily we select $ nk + 1$ chords among the points on the circle. Prove that of these chords there are at least $ k + 1$ chords which pairwise do not have a point in common.
2001 Irish Math Olympiad, 2
Three hoops are arranged concentrically as in the diagram. Each hoop is threaded with $ 20$ beads, $ 10$ of which are black and $ 10$ are white. On each hoop the positions of the beads are labelled $ 1$ through $ 20$ as shown. We say there is a match at position $ i$ if all three beads at position $ i$ have the same color. We are free to slide beads around a hoop, not breaking the hoop. Show that it is always possible to move them into a configuration involving no less than $ 5$ matches.
1989 IMO Longlists, 80
A balance has a left pan, a right pan, and a pointer that moves along a graduated ruler. Like many other grocer balances, this one works as follows: An object of weight $ L$ is placed in the left pan and another of weight $ R$ in the right pan, the pointer stops at the number $ R \minus{} L$ on the graduated ruler. There are $ n, (n \geq 2)$ bags of coins, each containing $ \frac{n(n\minus{}1)}{2} \plus{} 1$ coins. All coins look the same (shape, color, and so on). $ n\minus{}1$ bags contain real coins, all with the same weight. The other bag (we don’t know which one it is) contains false coins. All false coins have the same weight, and this weight is different from the weight of the real coins. A legal weighing consists of placing a certain number of coins in one of the pans, putting a certain number of coins in the other pan, and reading the number given by the pointer in the graduated ruler. With just two legal weighings it is possible to identify the bag containing false coins. Find a way to do this and explain it.
2002 Vietnam Team Selection Test, 2
On a blackboard a positive integer $n_0$ is written. Two players, $A$ and $B$ are playing a game, which respects the following rules:
$-$ acting alternatively per turn, each player deletes the number written on the blackboard $n_k$ and writes instead one number denoted with $n_{k+1}$ from the set $\left\{n_k-1, \dsp \left\lfloor\frac {n_k}3\right\rfloor\right\}$;
$-$ player $A$ starts first deleting $n_0$ and replacing it with $n_1\in\left\{n_0-1, \dsp \left\lfloor\frac {n_0}3\right\rfloor\right\}$;
$-$ the game ends when the number on the table is 0 - and the player who wrote it is the winner.
Find which player has a winning strategy in each of the following cases:
a) $n_0=120$;
b) $n_0=\dsp \frac {3^{2002}-1}2$;
c) $n_0=\dsp \frac{3^{2002}+1}2$.
1998 All-Russian Olympiad, 3
A set $\mathcal S$ of translates of an equilateral triangle is given in the plane, and any two have nonempty intersection. Prove that there exist three points such that every triangle in $\mathcal S$ contains one of these points.
1996 South africa National Olympiad, 4
In the Rainbow Nation there are two airways: Red Rockets and Blue Boeings. For any two cities in the Rainbow Nation it is possible to travel from the one to the other using either or both of the airways. It is known, however, that it is impossible to travel from Beanville to Mieliestad using only Red Rockets - not directly nor by travelling via other cities. Show that, using only Blue Boeings, one can travel from any city to any other city by stopping at at most one city along the way.
1997 Bulgaria National Olympiad, 3
Let $X$ be a set of $n + 1$ elements, $n\geq 2$. Ordered $n$-tuples $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$ formed from distinct elements of $X$ are called[i] disjoint [/i]if there exist distinct indices $1\leq i \neq j\leq n$ such that $a_i = b_j$. Find the maximal number of pairwise disjoint $n$-tuples.
2005 All-Russian Olympiad, 2
Given 2005 distinct numbers $a_1,\,a_2,\dots,a_{2005}$. By one question, we may take three different indices $1\le i<j<k\le 2005$ and find out the set of numbers $\{a_i,\,a_j,\,a_k\}$ (unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers $a_i$.
2011 China Girls Math Olympiad, 7
There are $n$ boxes ${B_1},{B_2},\ldots,{B_n}$ from left to right, and there are $n$ balls in these boxes. If there is at least $1$ ball in ${B_1}$, we can move one to ${B_2}$. If there is at least $1$ ball in ${B_n}$, we can move one to ${B_{n - 1}}$. If there are at least $2$ balls in ${B_k}$, $2 \leq k \leq n - 1$ we can move one to ${B_{k - 1}}$, and one to ${B_{k + 1}}$. Prove that, for any arrangement of the $n$ balls, we can achieve that each box has one ball in it.
2008 Tournament Of Towns, 2
Each of $4$ stones weights the integer number of grams. A balance with arrow indicates the di fference of weights on the left and the right sides of it. Is it possible to determine the weights of all stones in $4$ weighings, if the balance can make a mistake in $1$ gram in at most one weighing?
2009 Rioplatense Mathematical Olympiad, Level 3, 3
Alice and Bob play the following game. It begins with a set of $1000$ $1\times 2$ rectangles. A [i]move[/i] consists of choosing two rectangles (a rectangle may consist of one or several $1\times 2$ rectangles combined together) that share a common side length and combining those two rectangles into one rectangle along those sides sharing that common length. The first player who cannot make a move loses. Alice moves first. Describe a winning strategy for Bob.
2005 Lithuania Team Selection Test, 1
Find the smallest integer $n$ such that an $n\times n$ square can be partitioned into $40\times 40$ and $49\times 49$ squares, with both types of squares present in the partition, if
a) $40|n$; b) $49|n$; c) $n\in \mathbb N$.
2003 China Team Selection Test, 1
Let $S$ be the set of points inside and on the boarder of a regular haxagon with side length 1. Find the least constant $r$, such that there exists one way to colour all the points in $S$ with three colous so that the distance between any two points with same colour is less than $r$.
2002 All-Russian Olympiad, 2
We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$. Initially, the pack is situated on the red cell and arranged in an arbitrary order. In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell. Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?
2011 Postal Coaching, 2
For which $n \ge 1$ is it possible to place the numbers $1, 2, \ldots, n$ in some order $(a)$ on a line segment, or $(b)$ on a circle so that for every $s$ from $1$ to $\frac{n(n+1)}{2}$, there is a connected subset of the segement or circle such that the sum of the numbers in that subset is $s$?
2003 Indonesia MO, 6
The hall in a castle is a regular hexagon where its sides' length is 6 meters. The floor of the hall is to be tiled with equilateral triangular tiles where its sides' length is 50 centimeters. Each tile is divided into three congruent triangles by their altitudes up to its orthocenter (see below). Each of these small triangles are colored such that each tile has different colors and no two tiles have identical colorings. How many colors at least are required?
A tile's pattern is:
[asy]
draw((0,0.000)--(2,0.000));
draw((2,0.000)--(1,1.732));
draw((1,1.732)--(0,0.000));
draw((1,0.577)--(0,0.000));
draw((1,0.577)--(2,0.000));
draw((1,0.577)--(1,1.732));
[/asy]
2006 Rioplatense Mathematical Olympiad, Level 3, 3
The numbers $1, 2,\ldots, 2006$ are written around the circumference of a circle. A [i]move[/i] consists of exchanging two adjacent numbers. After a sequence of such moves, each number ends up $13$ positions to the right of its initial position. lf the numbers $1, 2,\ldots, 2006$ are partitioned into $1003$ distinct pairs, then show that in at least one of the moves, the two numbers of one of the pairs were exchanged.
2010 Contests, 4
In a football season, even number $n$ of teams plays a simple series, i.e. each team plays once against each other team. Show that ona can group the series into $n-1$ rounds such that in every round every team plays exactly one match.
2009 Argentina Iberoamerican TST, 3
Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.
2003 Baltic Way, 7
A subset of $X$ of $\{1,2,3, \ldots 10000 \}$ has the following property: If $a,b$ are distinct elements of $X$, then $ab\not\in X$. What is the maximal number of elements in $X$?
2006 MOP Homework, 7
Two concentric circles are divided by $n$ radii into $2n$ parts. Two parts are called neighbors (of each other) if they share either a
common side or a common arc. Initially, there are $4n + 1$ frogs inside the parts. At each second, if there are three or more frogs
inside one part, then three of the frogs in the part will jump to its neighbors, with one to each neighbor. Prove that in a finite
amount of time, for any part either there are frogs in the part or
there are frogs in each of its neighbors
1999 All-Russian Olympiad, 8
In a group of 12 persons, among any 9 there are 5 which know each other. Prove
that there are 6 persons in this group which know each other
2005 Olympic Revenge, 5
Find all sets X of points in a plane, not all collinear, such that:
For any two distinct circumferences, each contains three points of X, its intersection points are points of X.
2006 China Girls Math Olympiad, 6
Let $M= \{ 1, 2, \cdots, 19 \}$ and $A = \{ a_{1}, a_{2}, \cdots, a_{k}\}\subseteq M$. Find the least $k$ so that for any $b \in M$, there exist $a_{i}, a_{j}\in A$, satisfying $b=a_{i}$ or $b=a_{i}\pm a_{i}$ ($a_{i}$ and $a_{j}$ do not have to be different) .