Found problems: 1488
2011 Postal Coaching, 6
On a circle there are $n$ red and $n$ blue arcs given in such a way that each red arc intersects each blue one. Prove that some point is contained by at least $n$ of the given coloured arcs.
2007 Estonia Math Open Senior Contests, 9
Find all positive integers n such that one can write an integer 1 to $ n^2$ into each unit square of a $ n^2 \times n^2$ table in such a way that, in each row, each column and each $ n \times n$ block of unit squares, each number 1 to $ n^2$ occurs exactly once.
2000 All-Russian Olympiad, 8
Some paper squares of $k$ distinct colors are placed on a rectangular table, with sides parallel to the sides of the table. Suppose that for any $k$ squares of distinct colors, some two of them can be nailed on the table with only one nail. Prove that there is a color such that all squares of that color can be nailed with $2k-2$ nails.
2011 Turkey MO (2nd round), 6
Let $A$ and $B$ two countries which inlude exactly $2011$ cities.There is exactly one flight from a city of $A$ to a city of $B$ and there is no domestic flights (flights are bi-directional).For every city $X$ (doesn't matter from $A$ or from $B$), there exist at most $19$ different airline such that airline have a flight from $X$ to the another city.For an integer $k$, (it doesn't matter how flights arranged ) we can say that there exists at least $k$ cities such that it is possible to trip from one of these $k$ cities to another with same airline.So find the maximum value of $k$.
1974 IMO Longlists, 51
There are $n$ points on a flat piece of paper, any two of them at a distance of at least $2$ from each other. An inattentive pupil spills ink on a part of the paper such that the total area of the damaged part equals $\frac 32$. Prove that there exist two vectors of equal length less than $1$ and with their sum having a given direction, such that after a translation by either of these two vectors no points of the given set remain in the damaged area.
1991 China Team Selection Test, 3
$5$ points are given in the plane, any three non-collinear and any four non-concyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be [i]good[/i]. Let the number of good circles be $n$; find all possible values of $n$.
2005 China Girls Math Olympiad, 4
Determine all positive real numbers $ a$ such that there exists a positive integer $ n$ and sets $ A_1, A_2, \ldots, A_n$ satisfying the following conditions:
(1) every set $ A_i$ has infinitely many elements;
(2) every pair of distinct sets $ A_i$ and $ A_j$ do not share any common element
(3) the union of sets $ A_1, A_2, \ldots, A_n$ is the set of all integers;
(4) for every set $ A_i,$ the positive difference of any pair of elements in $ A_i$ is at least $ a^i.$
1991 Vietnam National Olympiad, 1
$1991$ students sit around a circle and play the following game. Starting from some student $A$ and counting clockwise, each student on turn says a number. The numbers are $1,2,3,1,2,3,...$ A student who says $2$ or $3$ must leave the circle. The game is over when there is only one student left. What position was the remaining student sitting at the beginning of the game?
1993 Vietnam Team Selection Test, 3
Let $n$ points $A_1, A_2, \ldots, A_n$, ($n>2$), be considered in the space, where no four points are coplanar. Each pair of points $A_i, A_j$ are connected by an edge. Find the maximal value of $n$ for which we can paint all edges by two colors – blue and red such that the following three conditions hold:
[b]I.[/b] Each edge is painted by exactly one color.
[b]II.[/b] For $i = 1, 2, \ldots, n$, the number of blue edges with one end $A_i$ does not exceed 4.
[b]III.[/b] For every red edge $A_iA_j$, we can find at least one point $A_k$ ($k \neq i, j$) such that the edges $A_iA_k$ and $A_jA_k$ are blue.
2001 Macedonia National Olympiad, 4
Let $\Omega$ be a family of subsets of $M$ such that:
$(\text{i})$ If $|A\cap B|\ge 2$ for $A,B\in\Omega$, then $A=B$;
$(\text{ii})$ There exist different subsets $A,B,C\in\Omega$ with $|A\cap B\cap C|=1$;
$(\text{iii})$ For every $A\in\Omega$ and $a\in M \ A$, there is a unique $B\in\Omega$ such that $a\in B$ and $A\cap B=\emptyset$.
Prove that there are numbers $p$ and $s$ such that:
$(1)$ Each $a\in M$ is contained in exactly $p$ sets in $\Omega$;
$(2)$ $|A|=s$ for all $A\in\Omega$;
$(3)$ $s+1\ge p$.
2002 Hong kong National Olympiad, 2
In conference there $n>2$ mathematicians. Every two mathematicians communicate in one of the $n$ offical languages of the conference. For any three different offical languages the exists three mathematicians who communicate with each other in these three languages. Find all $n$ such that this is possible.
2010 Mexico National Olympiad, 2
In each cell of an $n\times n$ board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off).
Show that if after a certain number of moves, at least one light is on, then at this moment at least $n$ lights are on.
2009 Croatia Team Selection Test, 2
On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.
1989 IMO Longlists, 36
Connecting the vertices of a regular $ n$-gon we obtain a closed (not necessarily convex) $ n$-gon. Show that if $ n$ is even, then there are two parallel segments among the connecting segments and if $ n$ is odd then there cannot be exactly two parallel segments.
2010 China Team Selection Test, 2
In a football league, there are $n\geq 6$ teams. Each team has a homecourt jersey and a road jersey with different color. When two teams play, the home team always wear homecourt jersey and the road team wear their homecourt jersey if the color is different from the home team's homecourt jersey, or otherwise the road team shall wear their road jersey. It is required that in any two games with 4 different teams, the 4 teams' jerseys have at least 3 different color. Find the least number of color that the $n$ teams' $2n$ jerseys may use.
2010 Serbia National Math Olympiad, 2
An $n\times n$ table whose cells are numerated with numbers $1, 2,\cdots, n^2$ in some order is called [i]Naissus[/i] if all products of $n$ numbers written in $n$ [i]scattered[/i] cells give the same residue when divided by $n^2+1$. Does there exist a Naissus table for
$(a) n = 8;$
$(b) n = 10?$
($n$ cells are [i]scattered[/i] if no two are in the same row or column.)
[i]Proposed by Marko Djikic[/i]
1988 IMO Longlists, 81
There are $ n \geq 3$ job openings at a factory, ranked $1$ to $ n$ in order of increasing pay. There are $ n$ job applicants, ranked from $1$ to $ n$ in order of increasing ability. Applicant $ i$ is qualified for job $ j$ if and only if $ i \geq j.$ The applicants arrive one at a time in random order. Each in turn is hired to the highest-ranking job for which he or she is qualified AND which is lower in rank than any job already filled. (Under these rules, job $1$ is always filled, and hiring terminates thereafter.) Show that applicants $ n$ and $ n \minus{} 1$ have the same probability of being hired.
2019 Serbia National MO, 5
In the spherical shaped planet $X$ there are $2n$ gas stations. Every station is paired with one other station ,
and every two paired stations are diametrically opposite points on the planet.
Each station has a given amount of gas. It is known that : if a car with empty (large enough) tank starting
from any station it is always to reach the paired station with the initial station (it can get extra gas during the journey).
Find all naturals $n$ such that for any placement of $2n$ stations for wich holds the above condotions, holds:
there always a gas station wich the car can start with empty tank and go to all other stations on the planet.(Consider that the car consumes a constant amount of gas per unit length.)
2006 Bulgaria Team Selection Test, 3
[b] Problem 6.[/b] Let $m\geq 5$ and $n$ are given natural numbers, and $M$ is regular $2n+1$-gon. Find the number of the convex $m$-gons with vertices among the vertices of $M$, who have at least one acute angle.
[i]Alexandar Ivanov[/i]
2001 Finnish National High School Mathematics Competition, 4
A sequence of seven digits is randomly chosen in a weekly lottery. Every digit can be any of the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$
What is the probability of having at most fi ve diff erent digits in the sequence?
1989 Federal Competition For Advanced Students, P2, 3
Show that it is possible to situate eight parallel planes at equal distances such that each plane contains precisely one vertex of a given cube. How many such configurations of planes are there?
2007 Tournament Of Towns, 7
For each letter in the English alphabet, William assigns an English word which contains that letter. His first document consists only of the word assigned to the letter $A$. In each subsequent document, he replaces each letter of the preceding document by its assigned word. The fortieth document begins with “Till whatsoever star that guides my moving.” Prove that this sentence reappears later in this document.
2008 Middle European Mathematical Olympiad, 2
On a blackboard there are $ n \geq 2, n \in \mathbb{Z}^{\plus{}}$ numbers. In each step we select two numbers from the blackboard and replace both of them by their sum. Determine all numbers $ n$ for which it is possible to yield $ n$ identical number after a finite number of steps.
1993 Polish MO Finals, 1
Let be given a convex polyhedron whose all faces are triangular. The vertices of the polyhedron are colored using three colors. Prove that the number of faces with vertices in all the three colors is even.
1982 IMO Longlists, 12
Let there be $3399$ numbers arbitrarily chosen among the first $6798$ integers $1, 2, \ldots , 6798$ in such a way that none of them divides another. Prove that there are exactly $1982$ numbers in $\{1, 2, \ldots, 6798\}$ that must end up being chosen.