Found problems: 1800
2002 Romania Team Selection Test, 4
For any positive integer $n$, let $f(n)$ be the number of possible choices of signs $+\ \text{or}\ - $ in the algebraic expression $\pm 1\pm 2\ldots \pm n$, such that the obtained sum is zero. Show that $f(n)$ satisfies the following conditions:
a) $f(n)=0$ for $n=1\pmod{4}$ or $n=2\pmod{4}$.
b) $2^{\frac{n}{2}-1}\le f(n)\le 2^n-2^{\lfloor\frac{n}{2}\rfloor+1}$, for $n=0\pmod{4}$ or $n=3\pmod{4}$.
[i]Ioan Tomsecu[/i]
1990 IMO Longlists, 55
Given points $A, M, M_1$ and rational number $\lambda \neq -1$. Construct the triangle $ABC$, such that $M$ lies on $BC$ and $M_1$ lies on $B_1C_1$ ($B_1, C_1$ are the projections of $B, C$ on $AC, AB$ respectively), and $\frac{BM}{MC}=\frac{B_1M_1}{M_1C_1}=\lambda.$
2021 Turkey Junior National Olympiad, 2
We are numbering the rows and columns of a $29 \text{x} 29$ chess table with numbers $1, 2, ..., 29$ in order (Top row is numbered with $1$ and first columns is numbered with $1$ as well). We choose some of the squares in this chess table and for every selected square, we know that there exist at most one square having a row number greater than or equal to this selected square's row number and a column number greater than or equal to this selected square's column number. How many squares can we choose at most?
2025 Bundeswettbewerb Mathematik, 4
For integers $m,n \ge 3$ we consider a $m \times n$ rectangular frame, consisting of the $2m+2n-4$ boundary squares of a $m \times n$ rectangle.
Renate and Erhard play the following game on this frame, with Renate to start the game. In a move, a player colours a rectangular area consisting of a single or several white squares. If there are any more white squares, they have to form a connected region. The player who moves last wins the game.
Determine all pairs $(m,n)$ for which Renate has a winning strategy.
2011 Canadian Mathematical Olympiad Qualification Repechage, 2
Brennan chooses a set $A = \{a, b,c, d, e \}$ of five real numbers with $a \leq b \leq c \leq d \leq e.$ Delaney determines the subsets of $A$ containing three numbers and adds up the numbers in these subsets. She obtains the sums $0, 3; 4, 8; 9, 10, 11, 12, 14, 19.$ What are the five numbers in Brennan's set?
2015 Turkey MO (2nd round), 3
$n$ points are given on a plane where $n\ge4$. All pairs of points are connected with a segment. Find the maximal number of segments which don't intersect with any other segments in their interior.
2012 Spain Mathematical Olympiad, 3
Let $x$ and $n$ be integers such that $1\le x\le n$. We have $x+1$ separate boxes and $n-x$ identical balls. Define $f(n,x)$ as the number of ways that the $n-x$ balls can be distributed into the $x+1$ boxes. Let $p$ be a prime number. Find the integers $n$ greater than $1$ such that the prime number $p$ is a divisor of $f(n,x)$ for all $x\in\{1,2,\ldots ,n-1\}$.
2007 Indonesia TST, 4
Let $ X$ be a set of $ k$ vertexes on a plane such that no three of them are collinear. Let $ P$ be the family of all $ {k \choose 2}$ segments that connect each pair of points. Determine $ \tau(P)$.
2014 Contests, 3
Let $N$ be an integer, $N>2$. Arnold and Bernold play the following game: there are initially $N$ tokens on a pile. Arnold plays first and removes $k$ tokens from the pile, $1\le k < N$. Then Bernold removes $m$ tokens from the pile, $1\le m\le 2k$ and so on, that is, each player, on its turn, removes a number of tokens from the pile that is between $1$ and twice the number of tokens his opponent took last. The player that removes the last token wins.
For each value of $N$, find which player has a winning strategy and describe it.
2005 Irish Math Olympiad, 4
Determine the number of arrangements $ a_1,a_2,...,a_{10}$ of the numbers $ 1,2,...,10$ such that $ a_i>a_{2i}$ for $ 1 \le i \le 5$ and $ a_i>a_{2i\plus{}1}$ for $ 1 \le i \le 4$.
2013 Tuymaada Olympiad, 1
$100$ heaps of stones lie on a table. Two players make moves in turn. At each move, a player can remove any non-zero number of stones from the table, so that at least one heap is left untouched. The player that cannot move loses. Determine, for each initial position, which of the players, the first or the second, has a winning strategy.
[i]K. Kokhas[/i]
[b]EDIT.[/b] It is indeed confirmed by the sender that empty heaps are still heaps, so the third post contains the right guess of an interpretation.
2005 Taiwan TST Round 2, 2
Starting from a positive integer $n$, we can replace the current number with a multiple of the current number or by deleting one or more zeroes from the decimal representation of the current number. Prove that for all values of $n$, it is possible to obtain a single-digit number by applying the above algorithm a finite number of times.
There is a nice solution to this...
2006 QEDMO 3rd, 7
Given a table with $2^n * n$ 1*1 squares ( $2^n$ rows and n column). In any square we put a number in {1, -1} such that no two rows are the same. Then we change numbers in some squares by 0. Prove that in new table we can choose some rows such that sum of all numbers in these rows equal to 0.
2005 Baltic Way, 8
Consider a $25 \times 25$ grid of unit squares. Draw with a red pen contours of squares of any size on the grid. What is the minimal number of squares we must draw in order to colour all the lines of the grid?
1991 Turkey Team Selection Test, 2
$p$ passengers get on a train with $n$ wagons. Find the probability of being at least one passenger at each wagon.
2008 Iran Team Selection Test, 7
Let $ S$ be a set with $ n$ elements, and $ F$ be a family of subsets of $ S$ with $ 2^{n\minus{}1}$ elements, such that for each $ A,B,C\in F$, $ A\cap B\cap C$ is not empty. Prove that the intersection of all of the elements of $ F$ is not empty.
1997 Turkey MO (2nd round), 3
Let $D_{1}, D_{2}, . . . , D_{n}$ be rectangular parallelepipeds in space, with edges parallel to the $x, y, z$ axes. For each $D_{i}$, let $x_{i} , y_{i} , z_{i}$ be the lengths of its projections onto the $x, y, z$ axes, respectively. Suppose that for all pairs $D_{i}$ , $D_{j}$, if at least one of $x_{i} < x_{j}$ , $y_{i} < y_{j}$, $z_{i} < z_{j}$ holds, then $x_{i} \leq x_{j}$ , $y_{i} \leq y_{j}$, and $z_{i} < z_{j}$ . If the volume of the region $\bigcup^{n}_{i=1}{D_{i}}$ equals 1997, prove that there is a subset $\{D_{i_{1}}, D_{i_{2}}, . . . , D_{i_{m}}\}$ of the set $\{D_{1}, . . . , D_{n}\}$ such that
$(i)$ if $k \not= l $ then $D_{i_{k}} \cap D_{i_{l}} = \emptyset $, and
$(ii)$ the volume of $\bigcup^{m}_{k=1}{D_{i_{k}}}$ is at least 73.
2007 USAMO, 4
An [i]animal[/i] with $n$ [i]cells[/i] is a connected figure consisting of $n$ equal-sized cells[1].
A [i]dinosaur[/i] is an animal with at least $2007$ cells. It is said to be [i]primitive[/i] it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
(1) Animals are also called [i]polyominoes[/i]. They can be defined inductively. Two cells are [i]adjacent[/i] if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
2009 Ukraine National Mathematical Olympiad, 4
Let $G$ be a connected graph, with degree of all vertices not less then $m \geq 3$, such that there is no path through all vertices of $G$ being in every vertex exactly once. Find the least possible number of vertices of $G.$
2013 Middle European Mathematical Olympiad, 2
Let $n$ be a positive integer. On a board consisting of $4n \times 4n$ squares, exactly $4n$ tokens are placed so that each row and each column contains one token. In a step, a token is moved horizontally of vertically to a neighbouring square. Several tokens may occupy the same square at the same time. The tokens are to be moved to occupy all the squares of one of the two diagonals.
Determine the smallest number $k(n)$ such that for any initial situation, we can do it in at most $k(n)$ steps.
1981 Miklós Schweitzer, 2
Consider the lattice $ L$ of the contradictions of a simple graph $ G$ (as sets of vertex pairs) with respect to inclusion. Let $ n \geq 1$ be an arbitrary integer. Show that the identity \[ x \bigwedge \left( \bigvee_{i\equal{}0}^n y_i \right) \equal{} \bigvee_{j\equal{}0}^n
\left( x \bigwedge \left( \bigvee_{0 \leq i \leq n, \;i\not\equal{}j\ } y_i \right)\right)\] holds if and only if $ G$ has no cycle of size at least $ n\plus{}2$.
[i]A. Huhn[/i]
2007 QEDMO 4th, 9
A team contest between $n$ participants is to be held. Each of these participants has exactly $k$ enemies among the other participants. (If $A$ is an enemy of $B$, then $B$ is an enemy of $A$. Nobody is his own enemy.) Assume that there are no three participants such that every two of them are enemies of each other.
A [i]subversive enmity[/i] will mean an (un-ordered) pair of two participants which are enemies of each other and which belong to one and the same team. Show that one can divide the participants into two teams such that the number of subversive enmities is $\leq\frac{k\left(n-2k\right)}{2}$. (The teams need not be of equal size.)
[i]Note.[/i] The actual source of this problem is:
Glenn Hopkins, William Staton, [i]Maximal Bipartite Subgraphs[/i], Ars Combinatoria 13 (1982), pp. 223-226.
It should be noticed that $\frac{k\left(n-2k\right)}{2}\leq\frac{n^{2}}{16}$, so the bound in the problem can be replaced by $\frac{n^{2}}{16}$ (which makes it weaker, but independent of $k$).
darij
2014 Mexico National Olympiad, 1
Each of the integers from 1 to 4027 has been colored either green or red. Changing the color of a number is making it red if it was green and making it green if it was red. Two positive integers $m$ and $n$ are said to be [i]cuates[/i] if either $\frac{m}{n}$ or $\frac{n}{m}$ is a prime number. A [i]step[/i] consists in choosing two numbers that are cuates and changing the color of each of them. Show it is possible to apply a sequence of steps such that every integer from 1 to 2014 is green.
2013 Cono Sur Olympiad, 3
[i]Nocycleland[/i] is a country with $500$ cities and $2013$ two-way roads, each one of them connecting two cities. A city $A$ [i]neighbors[/i] $B$ if there is one road that connects them, and a city $A$ [i]quasi-neighbors[/i] $B$ if there is a city $C$ such that $A$ neighbors $C$ and $C$ neighbors $B$.
It is known that in Nocycleland, there are no pair of cities connected directly with more than one road, and there are no four cities $A$, $B$, $C$ and $D$ such that $A$ neighbors $B$, $B$ neighbors $C$, $C$ neighbors $D$, and $D$ neighbors $A$.
Show that there is at least one city that quasi-neighbors at least $57$ other cities.
2005 Baltic Way, 7
A rectangular array has $ n$ rows and $ 6$ columns, where $ n \geq 2$. In each cell there is written either $ 0$ or $ 1$. All rows in the array are different from each other. For each two rows $ (x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})$ and $ (y_{1},y_{2},y_{3},y_{4},y_{5},y_{6})$, the row $ (x_{1}y_{1},x_{2}y_{2},x_{3}y_{3},x_{4}y_{4},x_{5}y_{5},x_{6}y_{6})$ can be found in the array as well. Prove that there is a column in which at least half of the entries are zeros.