Found problems: 1488
2014 Bosnia Herzegovina Team Selection Test, 2
It is given regular $n$-sided polygon, $n \geq 6$. How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon?
2001 Cono Sur Olympiad, 1
Each entry in a $2000\times 2000$ array is $0$, $1$, or $-1$. Show that it's possible for all $4000$ row sums and column sums to be distinct.
2010 Tournament Of Towns, 7
A square is divided into congruent rectangles with sides of integer lengths. A rectangle is important if it has at least one point in common with a given diagonal of the square. Prove that this diagonal bisects the total area of the important rectangles
2001 Federal Math Competition of S&M, Problem 3
Determine all positive integers $ n$ for which there is a coloring of all points in space so that each of the following conditions is satisfied:
(i) Each point is painted in exactly one color.
(ii) Exactly $ n$ colors are used.
(iii) Each line is painted in at most two different colors.
1989 IMO Longlists, 69
Let $ k$ and $ s$ be positive integers. For sets of real numbers $ \{\alpha_1, \alpha_2, \ldots , \alpha_s\}$ and $ \{\beta_1, \beta_2, \ldots, \beta_s\}$ that satisfy
\[ \sum^s_{i\equal{}1} \alpha^j_i \equal{} \sum^s_{i\equal{}1} \beta^j_i \quad \forall j \equal{} \{1,2 \ldots, k\}\]
we write \[ \{\alpha_1, \alpha_2, \ldots , \alpha_s\} \overset{k}{\equal{}} \{\beta_1, \beta_2, \ldots , \beta_s\}.\]
Prove that if \[ \{\alpha_1, \alpha_2, \ldots , \alpha_s\} \overset{k}{\equal{}} \{\beta_1, \beta_2, \ldots , \beta_s\}\] and $ s \leq k,$ then there exists a permutation $ \pi$ of $ \{1, 2, \ldots , s\}$ such that
\[ \beta_i \equal{} \alpha_{\pi(i)} \quad \forall i \equal{} 1,2, \ldots, s.\]
2005 MOP Homework, 7
Let $S$ be a set of points in the plane satisfying the following conditions:
(a) there are seven points in $S$ that form a convex heptagon; and
(b) for any five points in $S$, if they form a convex pentagon, then there is point in $S$ lies in the interior of the pentagon. Determine the minimum value of the number of elements in $S$.
2008 Tuymaada Olympiad, 1
Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly $ 5\over 4$ times greater than Vasya's result. What minimum number of portraits can be on the wall?
[i]Author: V. Frank[/i]
1988 IMO Longlists, 12
Show that there do not exist more than $27$ half-lines (or rays) emanating from the origin in the $3$-dimensional space, such that the angle between each pair of rays is $\geq \frac{\pi}{4}$.
2014 Czech and Slovak Olympiad III A, 3
Suppose we have a $8\times8$ chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into $1\times2$ domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers.
(Day 1, 3rd problem
author: Michal Rolínek)
2008 South East Mathematical Olympiad, 4
Let $m, n$ be positive integers $(m, n>=2)$. Given an $n$-element set $A$ of integers $(A=\{a_1,a_2,\cdots ,a_n\})$, for each pair of elements $a_i, a_j(j>i)$, we make a difference by $a_j-a_i$. All these $C^2_n$ differences form an ascending sequence called “derived sequence” of set $A$. Let $\bar{A}$ denote the derived sequence of set $A$. Let $\bar{A}(m)$ denote the number of terms divisible by $m$ in $\bar{A}$ . Prove that $\bar{A}(m)\ge \bar{B}(m)$ where $A=\{a_1,a_2,\cdots ,a_n\}$ and $B=\{1,2,\cdots ,n\}$.
2006 France Team Selection Test, 3
Let $M=\{1,2,\ldots,3 \cdot n\}$. Partition $M$ into three sets $A,B,C$ which $card$ $A$ $=$ $card$ $B$ $=$ $card$ $C$ $=$ $n .$
Prove that there exists $a$ in $A,b$ in $B, c$ in $C$ such that or $a=b+c,$ or $b=c+a,$ or $c=a+b$
[i]Edited by orl.[/i]
1978 Austrian-Polish Competition, 5
We are given $1978$ sets of size $40$ each. The size of the intersection of any two sets is exactly $1$. Prove that all the sets have a common element.
2012 Brazil National Olympiad, 5
In how many ways we can paint a $N \times N$ chessboard using 4 colours such that squares with a common side are painted with distinct colors and every $2 \times 2$ square (formed with 4 squares in consecutive lines and columns) is painted with the four colors?
2007 Bundeswettbewerb Mathematik, 2
At the start of the game there are $ r$ red and $ g$ green pieces/stones on the table. Hojoo and Kestutis make moves in turn. Hojoo starts. The person due to make a move, chooses a colour and removes $ k$ pieces of this colour. The number $ k$ has to be a divisor of the current number of stones of the other colour. The person removing the last piece wins. Who can force the victory?
2006 Bulgaria Team Selection Test, 1
[b]Problem 1. [/b]In the cells of square table are written the numbers $1$, $0$ or $-1$ so that in every line there is exactly one $1$, amd exactly one $-1$. Each turn we change the places of two columns or two rows. Is it possible, from any such table, after finite number of turns to obtain its opposite table (two tables are opposite if the sum of the numbers written in any two corresponding squares is zero)?
[i] Emil Kolev[/i]
2011 China Team Selection Test, 3
Let $G$ be a simple graph with $3n^2$ vertices ($n\geq 2$). It is known that the degree of each vertex of $G$ is not greater than $4n$, there exists at least a vertex of degree one, and between any two vertices, there is a path of length $\leq 3$. Prove that the minimum number of edges that $G$ might have is equal to $\frac{(7n^2- 3n)}{2}$.
1987 IMO Longlists, 72
Is it possible to cover a rectangle of dimensions $m \times n$ with bricks that have the trimino angular shape (an arrangement of three unit squares forming the letter $\text L$) if:
[b](a)[/b] $m \times n = 1985 \times 1987;$
[b](b)[/b] $m \times n = 1987 \times 1989 \quad ?$
2003 China Team Selection Test, 2
Suppose $A=\{1,2,\dots,2002\}$ and $M=\{1001,2003,3005\}$. $B$ is an non-empty subset of $A$. $B$ is called a $M$-free set if the sum of any two numbers in $B$ does not belong to $M$. If $A=A_1\cup A_2$, $A_1\cap A_2=\emptyset$ and $A_1,A_2$ are $M$-free sets, we call the ordered pair $(A_1,A_2)$ a $M$-partition of $A$. Find the number of $M$-partitions of $A$.
1998 Hungary-Israel Binational, 1
A player is playing the following game. In each turn he flips a coin and guesses the outcome. If his guess is correct, he gains $ 1$ point; otherwise he loses all his points. Initially the player has no points, and plays the game
until he has $ 2$ points.
(a) Find the probability $ p_{n}$ that the game ends after exactly $ n$ flips.
(b) What is the expected number of flips needed to finish the game?
2004 China Team Selection Test, 2
There are $ n \geq 5$ pairwise different points in the plane. For every point, there are just four points whose distance from which is $ 1$. Find the maximum value of $ n$.
1987 IMO Longlists, 49
In the coordinate system in the plane we consider a convex polygon $W$ and lines given by equations $x = k, y = m$, where $k$ and $m$ are integers. The lines determine a tiling of the plane with unit squares. We say that the boundary of $W$ intersects a square if the boundary contains an interior point of the square. Prove that the boundary of $W$ intersects at most $4 \lceil d \rceil $ unit squares, where $d$ is the maximal distance of points belonging to $W$ (i.e., the diameter of $W$) and $ \lceil d \rceil$ is the least integer not less than $d.$
1988 Kurschak Competition, 2
Set $T\subset\{1,2,\dots,n\}^3$ has the property that for any two triplets $(a,b,c)$ and $(x,y,z)$ in $T$, we have $a<b<c$, and also, we know that at most one of the equalities $a=x$, $b=y$, $c=z$ holds. Maximize $|T|$.
1997 Finnish National High School Mathematics Competition, 3
$12$ knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others.
$5$ knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice?
2011 India IMO Training Camp, 3
Consider a $ n\times n $ square grid which is divided into $ n^2 $ unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an $n$-staircase. Find the number of ways in which an $n$-stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.
2009 Croatia Team Selection Test, 2
Every natural number is coloured in one of the $ k$ colors. Prove that there exist four distinct natural numbers $ a, b, c, d$, all coloured in the same colour, such that $ ad \equal{} bc$, $ \displaystyle \frac b a$ is power of 2 and $ \displaystyle \frac c a$ is power of 3.