Found problems: 1800
1982 IMO Longlists, 3
Given $n$ points $X_1,X_2,\ldots, X_n$ in the interval $0 \leq X_i \leq 1, i = 1, 2,\ldots, n$, show that there is a point $y, 0 \leq y \leq 1$, such that
\[\frac{1}{n} \sum_{i=1}^{n} | y - X_i | = \frac 12.\]
1994 Baltic Way, 17
In a certain kingdom, the king has decided to build $25$ new towns on $13$ uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections.
2013 China Team Selection Test, 3
There are$n$ balls numbered $1,2,\cdots,n$, respectively. They are painted with $4$ colours, red, yellow, blue, and green, according to the following rules:
First, randomly line them on a circle.
Then let any three clockwise consecutive balls numbered $i, j, k$, in order.
1) If $i>j>k$, then the ball $j$ is painted in red;
2) If $i<j<k$, then the ball $j$ is painted in yellow;
3) If $i<j, k<j$, then the ball $j$ is painted in blue;
4) If $i>j, k>j$, then the ball $j$ is painted in green.
And now each permutation of the balls determine a painting method.
We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods.
Find out the number of all distinct painting methods.
2014 All-Russian Olympiad, 4
In a country of $n$ cities, an express train runs both ways between any two cities. For any train, ticket prices either direction are equal, but for any different routes these prices are different. Prove that the traveler can select the starting city, leave it and go on, successively, $n-1$ trains, such that each fare is smaller than that of the previous fare. (A traveler can enter the same city several times.)
1990 Flanders Math Olympiad, 3
We form a decimal code of $21$ digits. the code may start with $0$. Determine the probability that the fragment $0123456789$ appears in the code.
2014 Balkan MO, 4
Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length $1$ by lines parallel to its sides.
Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles.
[i]UK - Sahl Khan[/i]
2007 Ukraine Team Selection Test, 7
There are 25 people. Every two of them are use some language to speak between. They use only one language even if they both know another one. Among every three of them there is one who speaking with two other on the same language. Prove that there exist one who speaking with 10 other on the same language.
2003 China National Olympiad, 2
Determine the maximal size of the set $S$ such that:
i) all elements of $S$ are natural numbers not exceeding $100$;
ii) for any two elements $a,b$ in $S$, there exists $c$ in $S$ such that $(a,c)=(b,c)=1$;
iii) for any two elements $a,b$ in $S$, there exists $d$ in $S$ such that $(a,d)>1,(b,d)>1$.
[i]Yao Jiangang[/i]
1997 Baltic Way, 18
a) Prove the existence of two infinite sets $A$ and $B$, not necessarily disjoint, of non-negative integers such that each non-negative integer $n$ is uniquely representable in the form $n=a+b$ with $a\in A,b\in B$.
b) Prove that for each such pair $(A,B)$, either $A$ or $B$ contains only multiples of some integer $k>1$.
2014 IberoAmerican, 1
$N$ coins are placed on a table, $N - 1$ are genuine and have the same weight, and one is fake, with a different weight. Using a two pan balance, the goal is to determine with certainty the fake coin, and whether it is lighter or heavier than a genuine coin. Whenever one can deduce that one or more coins are genuine, they will be inmediately discarded and may no longer be used in subsequent weighings. Determine all $N$ for which the goal is achievable. (There are no limits regarding how many times one may use the balance).
Note: the only difference between genuine and fake coins is their weight; otherwise, they are identical.
2012 ELMO Shortlist, 4
A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles.
[i]Calvin Deng.[/i]
2004 Iran MO (3rd Round), 6
assume that we have a n*n table we fill it with 1,...,n such that each number exists exactly n times prove that there exist a row or column such that at least $\sqrt{n}$ diffrent number are contained.
2007 Iran Team Selection Test, 2
Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\frac{\pi}2]$. Find the largest value of the sum of the $\binom n2$ angles between line.
[i]By Aliakbar Daemi[/i]
2007 International Zhautykov Olympiad, 2
The set of positive nonzero real numbers are partitioned into three mutually disjoint non-empty subsets $(A\cup B\cup C)$.
a) show that there exists a triangle of side-lengths $a,b,c$, such that $a\in A, b\in B, c\in C$.
b) does it always happen that there exists a right triangle with the above property ?
2009 China Team Selection Test, 4
Let positive real numbers $ a,b$ satisfy $ b \minus{} a > 2.$ Prove that for any two distinct integers $ m,n$ belonging to $ [a,b),$ there always exists non-empty set $ S$ consisting of certain integers belonging to $ [ab,(a \plus{} 1)(b \plus{} 1))$ such that $ \frac {\displaystyle\prod_{x\in S}}{mn}$ is square of a rational number.
2010 All-Russian Olympiad, 1
There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors.
P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.
2008 Moldova Team Selection Test, 4
Find the number of even permutations of $ \{1,2,\ldots,n\}$ with no fixed points.
2013 All-Russian Olympiad, 1
$2n$ real numbers with a positive sum are aligned in a circle. For each of the numbers, we can see there are two sets of $n$ numbers such that this number is on the end. Prove that at least one of the numbers has a positive sum for both of these two sets.
1988 India National Olympiad, 3
Five men, $ A$, $ B$, $ C$, $ D$, $ E$ are wearing caps of black or white colour without each knowing the colour of his cap. It is known that a man wearing black cap always speaks the truth while the ones wearing white always tell lies. If they make the following statements, find the colour worn by each of them:
$ A$ : I see three black caps and one white cap.
$ B$ : I see four white caps
$ C$ : I see one black cap and three white caps
$ D$ : I see your four black caps.
2008 Tournament Of Towns, 1
Each of ten boxes contains a di fferent number of pencils. No two pencils in the same box are of the same colour. Prove that one can choose one pencil from each box so that no two are of the same colour.
1992 Turkey Team Selection Test, 2
There are $n$ boxes which is numbere from $1$ to $n$. The box with number $1$ is open, and the others are closed. There are $m$ identical balls ($m\geq n$). One of the balls is put into the open box, then we open the box with number $2$. Now, we put another ball to one of two open boxes, then we open the box with number $3$. Go on until the last box will be open. After that the remaining balls will be randomly put into the boxes. In how many ways this arrangement can be done?
2002 Tournament Of Towns, 1
There are many $a\times b$ rectangular cardboard pieces ($a,b\in\mathbb{N}$ such that $a<b$). It is given that by putting such pieces together without overlapping one can make $49\times 51$ rectangle, and $99\times 101$ rectangle. Can one uniquely determine $a,b$ from this?
1971 Miklós Schweitzer, 8
Show that the edges of a strongly connected bipolar graph can be oriented in such a way that for any edge $ e$ there is a simple directed path from pole $ p$ to pole $ q$ containing $ e$. (A strongly connected bipolar graph is a finite connected graph with two special vertices $ p$ and $ q$ having the property that there are no points $ x,y,x \not \equal{} y$, such that all paths from $ x$ to $ p$ as well as all paths from $ x$ to $ q$ contain $ y$.)
[i]A. Adam[/i]
2017 Baltic Way, 9
A positive integer $n$ is [i]Danish[/i] if a regular hexagon can be partitioned into $n$ congruent polygons. Prove that there are infinitely many positive integers $n$ such that both $n$ and $2^n+n$ are Danish.
2012 France Team Selection Test, 1
Let $n$ and $k$ be two positive integers. Consider a group of $k$ people such that, for each group of $n$ people, there is a $(n+1)$-th person that knows them all (if $A$ knows $B$ then $B$ knows $A$).
1) If $k=2n+1$, prove that there exists a person who knows all others.
2) If $k=2n+2$, give an example of such a group in which no-one knows all others.