Found problems: 1488
2008 Baltic Way, 15
Some $1\times 2$ dominoes, each covering two adjacent unit squares, are placed on a board of size $n\times n$ such that no two of them touch (not even at a corner). Given that the total area covered by the dominoes is $2008$, find the least possible value of $n$.
1987 IMO Longlists, 36
A game consists in pushing a flat stone along a sequence of squares $S_0, S_1, S_2, . . .$ that are arranged in linear order. The stone is initially placed on square $S_0$. When the stone stops on a square $S_k$ it is pushed again in the same direction and so on until it reaches $S_{1987}$ or goes beyond it; then the game stops. Each time the stone is pushed, the probability that it will advance exactly $n$ squares is $\frac{1}{2^n}$. Determine the probability that the stone will stop exactly on square $S_{1987}.$
1995 China Team Selection Test, 3
21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on?
1992 Hungary-Israel Binational, 3
We are given $100$ strictly increasing sequences of positive integers: $A_{i}= (a_{1}^{(i)}, a_{2}^{(i)},...), i = 1, 2,..., 100$. For $1 \leq r, s \leq 100$ we define the following quantities: $f_{r}(u)=$ the number of elements of $A_{r}$ not exceeding $n$; $f_{r,s}(u) =$ the number of elements of $A_{r}\cap A_{s}$ not exceeding $n$. Suppose that $f_{r}(n) \geq\frac{1}{2}n$ for all $r$ and $n$. Prove that there exists a pair of indices $(r, s)$ with $r \not = s$ such that $f_{r,s}(n) \geq\frac{8n}{33}$ for at least five distinct $n-s$ with $1 \leq n < 19920.$
2009 Croatia Team Selection Test, 2
In each field of 2009*2009 table you can write either 1 or -1.
Denote Ak multiple of all numbers in k-th row and Bj the multiple of all numbers in j-th column.
Is it possible to write the numbers in such a way that
$ \sum_{i\equal{}1}^{2009}{Ai}\plus{} \sum_{i\equal{}1}^{2009}{Bi}\equal{}0$?
2018 IFYM, Sozopol, 4
The cells of a table [b]m x n[/b], $m \geq 5$, $n \geq 5$ are colored in 3 colors where:
(i) Each cell has an equal number of adjacent (by side) cells from the other two colors;
(ii) Each of the cells in the 4 corners of the table doesn’t have an adjacent cell in the same color.
Find all possible values for $m$ and $n$.
2010 Czech-Polish-Slovak Match, 1
Given any collection of $2010$ nondegenerate triangles, their sides are painted so that each triangle has one red side, one blue side, and one white side. For each color, arrange the side lengths in order: [list][*]let $b_1\le b_2\le\cdots\le b_{2011}$ denote the lengths of the blue sides;
[*]let $r_1\le r_2\le\cdots\le r_{2011}$ denote the lengths of the red sides; and
[*]let $w_1\le w_2\le\cdots\le w_{2011}$ denote the lengths of the white sides.[/list] Find the largest integer $k$ for which there necessarily exists at least $k$ indices $j$ such that $b_j$, $r_j$, $w_j$ are the side lengths of a nondegenerate triangle.
2012 Kurschak Competition, 3
Consider $n$ events, each of which has probability $\frac12$. We also know that the probability of any two both happening is $\frac14$. Prove the following.
(a) The probability that none of these events happen is at most $\frac1{n+1}$.
(b) We can reach equality in (a) for infinitely many $n$.
1993 Baltic Way, 15
On each face of two dice some positive integer is written. The two dice are thrown and the numbers on the top face are added. Determine whether one can select the integers on the faces so that the possible sums are $2,3,4,5,6,7,8,9,10,11,12,13$, all equally likely?
1994 Kurschak Competition, 3
Consider the sets $A_1,A_2,\dots,A_n$. Set $A_k$ is composed of $k$ disjoint intervals on the real axis ($k=1,2,\dots,n$). Prove that from the intervals contained by these sets, one can choose $\left\lfloor\frac{n+1}2\right\rfloor$ intervals such that they belong to pairwise different sets $A_k$, and no two of these intervals have a common point.
2013 Iran MO (3rd Round), 4
We have constructed a rhombus by attaching two equal equilateral triangles. By putting $n-1$ points on all 3 sides of each triangle we have divided the sides to $n$ equal segments. By drawing line segements between correspounding points on each side of the triangles we have divided the rhombus into $2n^2$ equal triangles.
We write the numbers $1,2,\dots,2n^2$ on these triangles in a way no number appears twice. On the common segment of each two triangles we write the positive difference of the numbers written on those triangles. Find the maximum sum of all numbers written on the segments.
(25 points)
[i]Proposed by Amirali Moinfar[/i]
2013 Baltic Way, 7
A positive integer is written on a blackboard. Players $A$ and $B$ play the following game: in each move one has to choose a proper divisor $m$ of the number $n$ written on the blackboard ($1<m<n$) and replaces $n$ with $n-m$. Player $A$ makes the first move, then players move alternately. The player who can't make a move loses the game. For which starting numbers is there a winning strategy for player $B$?
1996 China Team Selection Test, 1
3 countries $A, B, C$ participate in a competition where each country has 9 representatives. The rules are as follows: every round of competition is between 1 competitor each from 2 countries. The winner plays in the next round, while the loser is knocked out. The remaining country will then send a representative to take on the winner of the previous round. The competition begins with $A$ and $B$ sending a competitor each. If all competitors from one country have been knocked out, the competition continues between the remaining 2 countries until another country is knocked out. The remaining team is the champion.
[b]I.[/b] At least how many games does the champion team win?
[b]II.[/b] If the champion team won 11 matches, at least how many matches were played?
1989 Irish Math Olympiad, 2
2. Each of $n$ members of a club is given a different item of information. The members are allowed to share the information, but, for security reasons, only in the following way: A pair may communicate by telephone. During a telephone call only one member may speak. The member who speaks may tell the other member all the information (s)he knows. Determine the minimal number of phone calls that are required to convey all the information to each of the members.
Hi, from my sketches I'm thinking the answer is $2n-2$ but I dont know how to prove that this number of calls is the smallest. Can anyone enlighten me? Thanks
2006 MOP Homework, 1
Determine all positive real numbers $a$ such that there exists a positive integer $n$ and partition $A_1$, $A_2$, ..., $A_n$ of infinity sets of the set of the integers satisfying the following condition: for every set $A_i$, the positive difference of any pair of elements in $A_i$ is at least $a^i$.
2009 Romanian Master of Mathematics, 2
A set $ S$ of points in space satisfies the property that all pairwise distances between points in $ S$ are distinct. Given that all points in $ S$ have integer coordinates $ (x,y,z)$ where $ 1 \leq x,y, z \leq n,$ show that the number of points in $ S$ is less than $ \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$
[i]Dan Schwarz, Romania[/i]
2010 Tournament Of Towns, 3
A $1\times 1\times 1$ cube is placed on an $8\times 8$ chessboard so that its bottom face coincides with a square of the chessboard. The cube rolls over a bottom edge so that the adjacent face now lands on the chessboard. In this way, the cube rolls around the chessboard, landing on each square at least once. Is it possible that a particular face of the cube never lands on the chessboard?
2007 Bosnia Herzegovina Team Selection Test, 6
The set $A$ has exactly $n>4$ elements. Ann chooses $n+1$ distinct subsets of $A$, such that every subset has exactly $3$ elements. Prove that there exist two subsets chosen by Ann which have exactly one common element.
2004 Turkey MO (2nd round), 6
Define $K(n,0)=\varnothing $ and, for all nonnegative integers m and n, $K(n,m+1)=\left\{ \left. k \right|\text{ }1\le k\le n\text{ and }K(k,m)\cap K(n-k,m)=\varnothing \right\}$. Find the number of elements of $K(2004,2004)$.
2015 All-Russian Olympiad, 6
A field has a shape of checkboard $\text{41x41}$ square. A tank concealed in one of the cells of the field. By one shot, a fighter airplane fires one of the cells. If a shot hits the tank, then the tank moves to a neighboring cell of the field, otherwise it stays in its cell (the cells are neighbours if they share a side). A pilot has no information about the tank , one needs to hit it twice. Find the least number of shots sufficient to destroy the tank for sure. [i](S.Berlov,A.Magazinov)[/i]
2007 Iran MO (2nd Round), 2
Two vertices of a cube are $A,O$ such that $AO$ is the diagonal of one its faces. A $n-$run is a sequence of $n+1$ vertices of the cube such that each $2$ consecutive vertices in the sequence are $2$ ends of one side of the cube. Is the $1386-$runs from $O$ to itself less than $1386-$runs from $O$ to $A$ or more than it?
2010 India National Olympiad, 4
How many 6-tuples $ (a_1,a_2,a_3,a_4,a_5,a_6)$ are there such that each of $ a_1,a_2,a_3,a_4,a_5,a_6$ is from the set $ \{1,2,3,4\}$ and the six expressions
\[ a_j^2 \minus{} a_ja_{j \plus{} 1} \plus{} a_{j \plus{} 1}^2\]
for $ j \equal{} 1,2,3,4,5,6$ (where $ a_7$ is to be taken as $ a_1$) are all equal to one another?
2010 Contests, 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.
2018 China Team Selection Test, 6
Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$, are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.$$ Now we know that $\left | S \right |\ge 2018,\, \left | T \right |\ge 2018.$ Try to find the minimal possible value of $n$.
2013 Baltic Way, 8
There are $n$ rooms in a sauna, each has unlimited capacity. No room may be attended by a female and a male simultaneously. Moreover, males want to share a room only with males that they don't know and females want to share a room only with females that they know. Find the biggest number $k$ such that any $k$ couples can visit the sauna at the same time, given that two males know each other if and only if their wives know each other.