Found problems: 1488
1976 IMO Longlists, 37
From a square board $11$ squares long and $11$ squares wide, the central square is removed. Prove that the remaining $120$ squares cannot be covered by $15$ strips each $8$ units long and one unit wide.
2016 Poland - Second Round, 6
$n$ ($n \ge 4$) green points are in a data space and no $4$ green points lie on one plane. Some segments which connect green points have been colored red. Number of red segments is even. Each two green points are connected with polyline which is build from red segments. Show that red segments can be split on pairs, such that segments from one pair have common end.
2013 Iran MO (3rd Round), 2
How many rooks can be placed in an $n\times n$ chessboard such that each rook is threatened by at most $2k$ rooks?
(15 points)
[i]Proposed by Mostafa Einollah zadeh[/i]
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].
1980 IMO, 17
Ten gamblers start playing with the same amount of money. In turn they cast five dice. If the dice show a total of $n$, the player must pay each other player $\frac{1}{n}$ times the sum which that player owns at the moment. They throw and pay one after the other. At the $10^{\text{th}}$ round (i.e. after each player has cast the five die once), the dice shows a total of $12$ and after the payment it turns out that every player has exactly the same sum as he had in the beginning. Is it possible to determine the totals shown by the dice at the nine former rounds?
2001 Tournament Of Towns, 4
On an east-west shipping lane are ten ships sailing individually. The first five from the west are sailing eastwards while the other five ships are sailing westwards. They sail at the same constant speed at all times. Whenever two ships meet, each turns around and sails in the opposite direction. When all ships have returned to port, how many meetings of two ships have taken place?
2009 Rioplatense Mathematical Olympiad, Level 3, 3
Call a permutation of the integers $(1,2,\ldots,n)$ [i]$d$-ordered[/i] if it does not contains a decreasing subsequence of length $d$. Prove that for every $d=2,3,\ldots,n$, the number of $d$-ordered permutations of $(1,2,\ldots,n)$ is at most $(d-1)^{2n}$.
2014 Kurschak Competition, 1
Consider a company of $n\ge 4$ people, where everyone knows at least one other person, but everyone knows at most $n-2$ of the others. Prove that we can sit four of these people at a round table such that all four of them know exactly one of their two neighbors. (Knowledge is mutual.)
2012 Baltic Way, 8
A directed graph does not contain directed cycles. The number of edges in any directed path does not exceed 99. Prove that it is possible to colour the edges of the graph in 2 colours so that the number of edges in any single-coloured directed path in the graph will not exceed 9.
1988 IMO Longlists, 51
The positive integer $n$ has the property that, in any set of $n$ integers, chosen from the integers $1,2, \ldots, 1988,$ twenty-nine of them form an arithmetic progression. Prove that $n > 1788.$
1971 IMO Longlists, 13
One Martian, one Venusian, and one Human reside on Pluton. One day they make the following conversation:
[b]Martian [/b]: I have spent $1/12$ of my life on Pluton.
[b]Human [/b]: I also have.
[b]Venusian [/b]: Me too.
[b]Martian [/b]: But Venusian and I have spend much more time here than you, Human.
[b]Human [/b]: That is true. However, Venusian and I are of the same age.
[b]Venusian [/b]: Yes, I have lived $300$ Earth years.
[b]Martian [/b]: Venusian and I have been on Pluton for the past $13$ years.
It is known that Human and Martian together have lived $104$ Earth years. Find the ages of Martian, Venusian, and Human.*
[hide="*"][i]*: Note that the numbers in the problem are not necessarily in base $10.$[/i][/hide]
2003 Tournament Of Towns, 5
Prior to the game John selects an integer greater than $100$.
Then Mary calls out an integer $d$ greater than $1$. If John's integer is divisible by $d$, then Mary wins. Otherwise, John subtracts $d$ from his number and the game continues (with the new number). Mary is not allowed to call out any number twice. When John's number becomes negative, Mary loses. Does Mary have a winning strategy?
2005 Korea National Olympiad, 8
A group of 6 students decided to make [i]study groups[/i] and [i]service activity groups[/i] according to the following principle:
Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.
Supposing there are at least one group and no three students belong to the same study group and service activity group, find the minimum number of groups.
2014 Cono Sur Olympiad, 1
Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$.
Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$
2002 Finnish National High School Mathematics Competition, 5
There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane.
The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of diff erent colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ including $A$ and $B.$
What is the least number of colours which suffices?
2004 Switzerland Team Selection Test, 5
A brick has the shape of a cube of size $2$ with one corner unit cube removed. Given a cube of side $2^{n}$ divided into unit cubes from which an arbitrary unit cube is removed, show that the remaining figure can be built using the described bricks.
2012 Brazil National Olympiad, 1
In a culturing of bacteria, there are two species of them: red and blue bacteria.
When two red bacteria meet, they transform into one blue bacterium.
When two blue bacteria meet, they transform into four red bacteria.
When a red and a blue bacteria meet, they transform into three red bacteria.
Find, in function of the amount of blue bacteria and the red bacteria initially in the culturing,
all possible amounts of bacteria, and for every possible amount, the possible amounts of red and blue bacteria.
1982 USAMO, 1
In a party with $1982$ persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?
2009 Tournament Of Towns, 5
A country has two capitals and several towns. Some of them are connected by roads. Some of the roads are toll roads where a fee is charged for driving along them. It is known that any route from the south capital to the north capital contains at least ten toll roads. Prove that all toll roads can be distributed among ten companies so that anybody driving from the south capital to the north capital must pay each of these companies.
[i](5 points)[/i]
2000 All-Russian Olympiad, 5
Let $M$ be a finite sum of numbers, such that among any three of its elements there are two whose sum belongs to $M$. Find the greatest possible number of elements of $M$.
1991 China Team Selection Test, 3
All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.
2008 Tuymaada Olympiad, 6
A set $ X$ of positive integers is called [i]nice[/i] if for each pair $ a$, $ b\in X$ exactly one of the numbers $ a \plus{} b$ and $ |a \minus{} b|$ belongs to $ X$ (the numbers $ a$ and $ b$ may be equal). Determine the number of nice sets containing the number 2008.
[i]Author: Fedor Petrov[/i]
2010 Contests, 3
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
1990 China Team Selection Test, 4
There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.
2009 Albania Team Selection Test, 3
Two people play a game as follows: At the beginning both of them have one point and in every move, one of them can double it's points, or when the other have more point than him, subtract to him his points. Can the two competitors have 2009 and 2002 points respectively? What about 2009 and 2003? Generally which couples of points can they have?