Found problems: 1800
1996 Baltic Way, 19
Four heaps contain $38,45,61$ and $70$ matches respectively. Two players take turn choosing any two of the heaps and take some non-zero number of matches from one heap and some non-zero number of matches from the other heap. The player who cannot make a move, loses. Which one of the players has a winning strategy ?
2014 International Zhautykov Olympiad, 3
Given are 100 different positive integers. We call a pair of numbers [i]good[/i] if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.)
[i]Proposed by Alexander S. Golovanov, Russia[/i]
1970 IMO Longlists, 57
Let the numbers $1, 2, \ldots , n^2$ be written in the cells of an $n \times n$ square board so that the entries in each column are arranged increasingly. What are the smallest and greatest possible sums of the numbers in the $k^{th}$ row? ($k$ a positive integer, $1 \leq k \leq n$.)
2014 CentroAmerican, 1
Using squares of side 1, a stair-like figure is formed in stages following the pattern of the drawing.
For example, the first stage uses 1 square, the second uses 5, etc. Determine the last stage for which the corresponding figure uses less than 2014 squares.
[img]http://www.artofproblemsolving.com/Forum/download/file.php?id=49934[/img]
2001 Bulgaria National Olympiad, 1
Let $n \geq 2$ be a given integer. At any point $(i, j)$ with $i, j \in\mathbb{ Z}$ we write the remainder of $i+j$ modulo $n$. Find all pairs $(a, b)$ of positive integers such that the rectangle with vertices $(0, 0)$, $(a, 0)$, $(a, b)$, $(0, b)$ has the following properties:
[b](i)[/b] the remainders $0, 1, \ldots , n-1$ written at its interior points appear the same number of times;
[b](ii)[/b] the remainders $0, 1, \ldots , n -1$ written at its boundary points appear the same number of times.
1971 IMO Longlists, 22
We are given an $n \times n$ board, where $n$ is an odd number. In each cell of the board either $+1$ or $-1$ is written. Let $a_k$ and $b_k$ denote them products of numbers in the $k^{th}$ row and in the $k^{th}$ column respectively. Prove that the sum $a_1 +a_2 +\cdots+a_n +b_1 +b_2 +\cdots+b_n$ cannot be equal to zero.
2010 Contests, 1
There are ten coins a line, which are indistinguishable. It is known that two of them are false and have consecutive positions on the line. For each set of positions, you may ask how many false coins it contains. Is it possible to identify the false coins by making only two of those questions, without knowing the answer to the first question before making the second?
2010 Indonesia TST, 1
The integers $ 1,2,\dots,20$ are written on the blackboard. Consider the following operation as one step: [i]choose two integers $ a$ and $ b$ such that $ a\minus{}b \ge 2$ and replace them with $ a\minus{}1$ and $ b\plus{}1$[/i]. Please, determine the maximum number of steps that can be done.
[i]Yudi Satria, Jakarta[/i]
2010 Contests, 1
A finite set of integers is called [i]bad[/i] if its elements add up to $2010$. A finite set of integers is a [i]Benelux-set[/i] if none of its subsets is bad. Determine the smallest positive integer $n$ such that the set $\{502, 503, 504, . . . , 2009\}$ can be partitioned into $n$ Benelux-sets.
(A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.)
[i](2nd Benelux Mathematical Olympiad 2010, Problem 1)[/i]
2004 Junior Balkan MO, 4
Consider a convex polygon having $n$ vertices, $n\geq 4$. We arbitrarily decompose the polygon into triangles having all the vertices among the vertices of the polygon, such that no two of the triangles have interior points in common. We paint in black the triangles that have two sides that are also sides of the polygon, in red if only one side of the triangle is also a side of the polygon and in white those triangles that have no sides that are sides of the polygon.
Prove that there are two more black triangles that white ones.
2010 Pan African, 2
How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?
2011 ELMO Shortlist, 1
Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that
a) If $A\subseteq S$, then $A\in F$ if and only if $S\setminus A\notin F$;
b) If $A\subseteq B\subseteq S$ and $B\in F$, then $A\in F$.
Determine if there must exist a function $f:S\to\mathbb{R}$ such that for every $A\subseteq S$, $A\in F$ if and only if
\[\sum_{s\in A}f(s)<\sum_{s\in S\setminus A}f(s).\]
[i]Evan O'Dorney.[/i]
2013 Brazil National Olympiad, 2
Arnaldo and Bernaldo play the following game: given a fixed finite set of positive integers $A$ known by both players, Arnaldo picks a number $a \in A$ but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer $b$ (not necessarily in $A$). Then Arnaldo tells the number of divisors of $ab$. Show that Bernaldo can choose $b$ in a way that he can find out the number $a$ chosen by Arnaldo.
1995 All-Russian Olympiad, 6
A boy goes $n$ times at a merry-go-round with $n$ seats. After every time he moves in the clockwise direction and takes another seat, not making a full circle. The number of seats he passes by at each move is called the length of the move. For which $n$ can he sit at every seat, if the lengths of all the $n-1$ moves he makes have different lengths?
[i]V. New[/i]
2007 Tuymaada Olympiad, 1
What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?
2006 All-Russian Olympiad, 7
A $100\times 100$ chessboard is cut into dominoes ($1\times 2$ rectangles). Two persons play the following game: At each turn, a player glues together two adjacent cells (which were formerly separated by a cut-edge). A player loses if, after his turn, the $100\times 100$ chessboard becomes connected, i. e. between any two cells there exists a way which doesn't intersect any cut-edge. Which player has a winning strategy - the starting player or his opponent?
2006 Cono Sur Olympiad, 6
We divide the plane in squares shaped of side 1, tracing straight lines parallel bars to the coordinate axles. Each square is painted of black white or. To each as, we recolor all simultaneously squares, in accordance with the following rule: each square $Q$ adopts the color that more appears in the
configuration of five squares indicated in the figure. The recoloration process is repeated indefinitely.
Determine if exists an initial coloration with black a finite amount of squares such that always has at least one black square, not mattering how many seconds if had passed since the beginning of the process.
2012 All-Russian Olympiad, 4
Initially there are $n+1$ monomials on the blackboard: $1,x,x^2, \ldots, x^n $. Every minute each of $k$ boys simultaneously write on the blackboard the sum of some two polynomials that were written before. After $m$ minutes among others there are the polynomials $S_1=1+x,S_2=1+x+x^2,S_3=1+x+x^2+x^3,\ldots ,S_n=1+x+x^2+ \ldots +x^n$ on the blackboard. Prove that $ m\geq \frac{2n}{k+1} $.
2011 South East Mathematical Olympiad, 3
Find all positive integer $n$ , such that for all 35-element-subsets of $M=(1,2,3,...,50)$ ,there exists at least two different elements $a,b$ , satisfing : $a-b=n$ or $a+b=n$.
2009 China Team Selection Test, 1
Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$
2012 Iran MO (3rd Round), 4
We have $n$ bags each having $100$ coins. All of the bags have $10$ gram coins except one of them which has $9$ gram coins. We have a balance which can show weights of things that have weight of at most $1$ kilogram. At least how many times shall we use the balance in order to find the different bag?
[i]Proposed By Hamidreza Ziarati[/i]
2007 Kazakhstan National Olympiad, 4
Several identical square sheets of paper are laid out on a rectangular table so that their sides are parallel to the edges of the table (sheets may overlap). Prove that you can stick a few pins in such a way that each sheet will be attached to the table exactly by one pin.
2005 Junior Tuymaada Olympiad, 4
The organizers of a mathematical congress found that if they accomodate any participant in a room the rest can be accomodated in double rooms so that 2 persons living in each room know each other. Prove that every participant can organize a round table on graph theory for himself and an even number of other people so that each participant of the round table knows both his neigbours.
[i]Proposed by S. Berlov, S. Ivanov[/i]
2008 China Team Selection Test, 2
In a plane, there is an infinite triangular grid consists of equilateral triangles whose lengths of the sides are equal to $ 1$, call the vertices of the triangles the lattice points, call two lattice points are adjacent if the distance between the two points is equal to $ 1;$
A jump game is played by two frogs $ A,B,$ "A jump" is called if the frogs jump from the point which it is lying on to its adjacent point, " A round jump of $ A,B$" is called if first $ A$ jumps and then $ B$ by the following rules:
Rule (1): $ A$ jumps once arbitrarily, then $ B$ jumps once in the same direction, or twice in the opposite direction;
Rule (2): when $ A,B$ sits on adjacent lattice points, they carry out Rule (1) finishing a round jump, or $ A$ jumps twice continually, keep adjacent with $ B$ every time, and $ B$ rests on previous position;
If the original positions of $ A,B$ are adjacent lattice points, determine whether for $ A$ and $ B$,such that the one can exactly land on the original position of the other after a finite round jumps.
2020 Turkey EGMO TST, 4
Every square of a $2020 \times 2020$ chess table is painted in red or white. For every two columns and two rows, at least two of the intersection squares satisfies that they are in the same column or row and they are painted in the same color. Find the least value of number of columns and rows that are completely painted in one color.