Found problems: 1488
1987 China Team Selection Test, 3
Let $ G$ be a simple graph with $ 2 \cdot n$ vertices and $ n^{2}+1$ edges. Show that this graph $ G$ contains a $ K_{4}-\text{one edge}$, that is, two triangles with a common edge.
2013 China Western Mathematical Olympiad, 7
Label sides of a regular $n$-gon in clockwise direction in order 1,2,..,n. Determine all integers n ($n\geq 4$) satisfying the following conditions:
(1) $n-3$ non-intersecting diagonals in the $n$-gon are selected, which subdivide the $n$-gon into $n-2$ non-overlapping triangles;
(2) each of the chosen $n-3$ diagonals are labeled with an integer, such that the sum of labeled numbers on three sides of each triangles in (1) is equal to the others;
1970 IMO Longlists, 41
Let a cube of side $1$ be given. Prove that there exists a point $A$ on the surface $S$ of the cube such that every point of $S$ can be joined to $A$ by a path on $S$ of length not exceeding $2$. Also prove that there is a point of $S$ that cannot be joined with $A$ by a path on $S$ of length less than $2$.
2001 German National Olympiad, 2
Determine the maximum possible number of points you can place in a rectangle with lengths $14$ and $28$ such that any two of those points are more than $10$ apart from each other.
1988 IMO Longlists, 73
A two-person game is played with nine boxes arranged in a $3 \times 3$ square and with white and black stones. At each move a player puts three stones, not necessarily of the same colour, in three boxes in either a horizontal or a vertical line. No box can contain stones of different colours: if, for instance, a player puts a white stone in a box containing black stones the white stone and one of the black stones are removed from the box. The game is over when the centrebox and the cornerboxes contain one black stone and the other boxes are empty. At one stage of a game $x$ boxes contained one black stone each and the other boxes were empty. Determine all possible values for $x.$
2000 Italy TST, 4
On a mathematical competition $ n$ problems were given. The final results showed that:
(i) on each problem, exactly three contestants scored $ 7$ points;
(ii) for each pair of problems, exactly one contestant scored $ 7$ points on both problems.
Prove that if $ n \geq 8$, then there is a contestant who got $ 7$ points on each problem. Is this statement necessarily true if $ n \equal{} 7$?
1986 China Team Selection Test, 4
Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points.
[b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$.
[b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.
2001 All-Russian Olympiad, 1
Yura put $2001$ coins of $1$, $2$ or $3$ kopeykas in a row. It turned out that between any two $1$-kopeyka coins there is at least one coin; between any two $2$-kopeykas coins there are at least two coins; and between any two $3$-kopeykas coins there are at least $3$ coins. How many $3$-koyepkas coins could Yura put?
2008 Rioplatense Mathematical Olympiad, Level 3, 1
In each square of a chessboard with $a$ rows and $b$ columns, a $0$ or $1$ is written satisfying the following conditions.
[list][*]If a row and a column intersect in a square with a $0$, then that row and column have the same number of $0$s.
[*]If a row and a column intersect in a square with a $1$, then that row and column have the same number of $1$s.[/list]
Find all pairs $(a,b)$ for which this is possible.
2010 Postal Coaching, 3
In a group of $k$ people, some are acquainted with each other and some are not. Every evening, one person invites all his acquaintances to a party and introduces them to each other(if they have not already acquainted). Suppose that after each person has arranged at least one party, some two people do not know each other. Prove that they do not meet each other in the next party.
2014 Saudi Arabia BMO TST, 5
Let $n > 3$ be an odd positive integer not divisible by $3$. Determine if it is possible to form an $n \times n$ array of numbers such that
[list]
[*] [b](a)[/b] the set of the numbers in each row is a permutation of $0, 1, \dots , n - 1$;
the set of the numbers in each column is a permutation of $0, 1, \dots , n-1$;
[*] [b](b)[/b] the board is [i]totally non-symmetric[/i]: for $1 \le i < j \le n$ and $1 \le i' < j' \le n$, if $(i, j) \neq (i', j')$ then $(a_{i,j} , a_{j,i}) \neq (a_{i',j'} , a_{j',i'})$ where $a_{i,j}$ denotes the entry in the $i^\text{th}$ row and $j^\text{th}$ column.[/list]
2019 Latvia Baltic Way TST, 6
A grandpa has a finite number of boxes in his attic. Each box is a straight rectangular prism with integer edge lengths. For every box its width is greater or equal to its height and its length is greater or equal to its width. A box can be put inside another box if and only if all of its dimensions are respectively smaller than the other one's. You can put two or more boxes in a bigger box only if the smaller boxes are all already inside one of the boxes.
The grandpa decided to put the boxes in each other so that there would be a minimal number of visible boxes in the attic (boxes that have not been put inside another). He decided to use the following algorithm: at each step he finds the longest sequence of boxes so that the first can be put in the second, the second can be put in the third, etc., and then he puts them inside each other in the aforementioned order. The grandpa used the algorithm until no box could be put inside another. It is known that at each step the longest sequence of boxes was unique, e.g., at no moment were there two different sequences with the same length.
The grandpa now claims that he has the minimal possible number of visible boxes in his attic. Is the claim necessarily true?
2013 Indonesia MO, 8
Let $A$ be a set of positive integers. $A$ is called "balanced" if [and only if] the number of 3-element subsets of $A$ whose elements add up to a multiple of $3$ is equal to the number of 3-element subsets of $A$ whose elements add up to not a multiple of $3$.
a. Find a 9-element balanced set.
b. Prove that no set of $2013$ elements can be balanced.
2010 Tournament Of Towns, 3
At a circular track, $10$ cyclists started from some point at the same time in the same direction with different constant speeds. If any two cyclists are at some point at the same time again, we say that they meet. No three or more of them have met at the same time. Prove that by the time every two cyclists have met at least once, each cyclist has had at least $25$ meetings.
2014 Portugal MO, 3
Amélia and Beatriz play battleship on a $2n\times2n$ board, using very peculiar rules. Amélia begins by choosing $n$ lines and $n$ columns of the board, placing her $n^2$ submarines on the cells that lie on their intersections. Next, Beatriz chooses a set of cells that will explode. Which is the least number of cells that Beatriz has to choose in order to assure that at least a submarine will explode?
2003 Balkan MO, 4
A rectangle $ABCD$ has side lengths $AB = m$, $AD = n$, with $m$ and $n$ relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points $A_1 = A, A_2, A_3, \ldots , A_k = C$. Show that \[ A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}. \]
2007 Hungary-Israel Binational, 1
A given rectangle $ R$ is divided into $mn$ small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of $ R$?
2012 Baltic Way, 9
Zeroes are written in all cells of a $5 \times 5$ board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it. Is it possible to obtain the number 2012 in all cells simultaneously?
1989 India National Olympiad, 3
Let $ A$ denote a subset of the set $ \{ 1,11,21,31, \dots ,541,551 \}$ having the property that no two elements of $ A$ add up to $ 552$. Prove that $ A$ can't have more than $ 28$ elements.
1995 All-Russian Olympiad Regional Round, 10.8
The streets of the city of Duzhinsk are simple polygonal lines not intersecting each other in internal points. Each street connects two crossings and is colored in one of three colors: white, red, or blue. At each crossing exactly three streets meet, one of each color. A crossing is called positive if the streets meeting at it are white, blue and red in counterclockwise
direction, and negative otherwise. Prove that the difference between the numbers of positive and negative crossings is a multiple of 4.
2004 India IMO Training Camp, 3
Two runners start running along a circular track of unit length from the same starting point and int he same sense, with constant speeds $v_1$ and $v_2$ respectively, where $v_1$ and $v_2$ are two distinct relatively prime natural numbers. They continue running till they simultneously reach the starting point. Prove that
(a) at any given time $t$, at least one of the runners is at a distance not more than $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units from the starting point.
(b) there is a time $t$ such that both the runners are at least $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units away from the starting point. (All disstances are measured along the track). $[x]$ is the greatest integer function.
2004 China Western Mathematical Olympiad, 2
All the grids of a $m\times n$ chess board ($m,n\geq 3$), are colored either with red or with blue. Two adjacent grids (having a common side) are called a "good couple" if they have different colors. Suppose there are $S$ "good couples". Explain how to determine whether $S$ is odd or even. Is it prescribed by
some specific color grids? Justify your answers.
2006 China Team Selection Test, 3
$d$ and $n$ are positive integers such that $d \mid n$. The n-number sets $(x_1, x_2, \cdots x_n)$ satisfy the following condition:
(1) $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n$
(2) $d \mid (x_1+x_2+ \cdots x_n)$
Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy $x_n=n$.
1989 Polish MO Finals, 3
The edges of a cube are labeled from $1$ to $12$. Show that there must exist at least eight triples $(i, j, k)$ with $1 \leq i < j < k \leq 12$ so that the edges $i, j, k$ are consecutive edges of a path. Also show that there exists labeling in which we cannot find nine such triples.
1997 India National Olympiad, 5
Find the number of $4 \times 4$ array whose entries are from the set $\{ 0 , 1, 2, 3 \}$ and which are such that the sum of the numbers in each of the four rows and in each of the four columns is divisible by $4$.