Found problems: 1488
2007 Tournament Of Towns, 2
Initially, the number $1$ and two positive numbers $x$ and $y$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write on the blackboard, in a finite number of moves, the number
[list][b]a)[/b] $x^2$;
[b]b)[/b] $xy$?[/list]
2002 Olympic Revenge, 4
Find all pairs \((m,n)\) of positive integers such that there exists a polyhedron, with all faces being regular polygons, such that each vertex of the polyhedron is the vertex of exactly three faces, two of them having \(m\) sides, and the other having \(n\) sides.
2004 Korea National Olympiad, 4
Let $k$ and $N$ be positive real numbers which satisfy $k\leq N$. For $1\leq i \leq k$, there are subsets $A_i$ of $\{1,2,3,\ldots,N\}$ that satisfy the following property.
For arbitrary subset of $\{ i_1, i_2, \ldots , i_s \} \subset \{ 1, 2, 3, \ldots, k \} $, $A_{i_1} \triangle A_{i_2} \triangle ... \triangle A_{i_s}$ is not an empty set.
Show that a subset $\{ j_1, j_2, .. ,j_t \} \subset \{ 1, 2, ... ,k \} $ exist that satisfies $n(A_{j_1} \triangle A_{j_2} \triangle \cdots \triangle A_{j_t}) \geq k$. ($A \triangle B=A \cup B-A \cap B$)
2001 All-Russian Olympiad, 4
Consider a convex $2000$-gon, no three of whose diagonals have a common point. Each of its diagonals is colored in one of $999$ colors. Prove that there exists a triangle all of whose sides lie on diagonals of the same color. (Vertices of the triangle need not be vertices of the original polygon.)
1982 IMO Longlists, 42
Let $\mathfrak F$ be the family of all $k$-element subsets of the set $\{1, 2, \ldots, 2k + 1\}$. Prove that there exists a bijective function $f :\mathfrak F \to \mathfrak F$ such that for every $A \in \mathfrak F$, the sets $A$ and $f(A)$ are disjoint.
2008 Baltic Way, 13
For an upcoming international mathematics contest, the participating countries were asked to choose from nine combinatorics problems. Given how hard it usually is to agree, nobody was surprised that the following happened:
[b]i)[/b] Every country voted for exactly three problems.
[b]ii)[/b] Any two countries voted for different sets of problems.
[b]iii)[/b] Given any three countries, there was a problem none of them voted for.
Find the maximal possible number of participating countries.
1993 Turkey Team Selection Test, 4
Some towns are connected by roads, with at most one road between any two towns. Let $v$ be the number of towns and $e$ be the number of roads. Prove that
$(a)$ if $e<v-1$, then there are two towns such that one cannot travel between them;
$(b)$ if $2e>(v-1)(v-2)$, then one can travel between any two towns.
1991 Cono Sur Olympiad, 1
A game consists in $9$ coins (blacks or whites) arrenged in the following position (see picture 1). If you choose $1$ coin on the border of the square, this coin and it's neighbours change their color. If you choose the coin at the centre, it doesn't change it's color, but the other $8$ coins do. Here is an example of $9$ white coins, and the changes of their colors, choosing the coin said: (see picture 2).
Is it possible, starting with $9$ white coins, to have $9$ black coins?.
2002 Miklós Schweitzer, 2
Let $G$ be a simple $k$ edge-connected graph on $n$ vertices and let $u$ and $v$ be different vertices of $G$. Prove that there are $k$ edge-disjoint paths from $u$ to $v$ each having at most $\frac{20n}{k}$ edges.
2014 Iran Team Selection Test, 4
Find the maximum number of Permutation of set {$1,2,3,...,2014$} such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation
$b$ comes exactly after $a$
2010 Germany Team Selection Test, 1
Consider 2009 cards which are lying in sequence on a table. Initially, all cards have their top face white and bottom face black. The cards are enumerated from 1 to 2009. Two players, Amir and Ercole, make alternating moves, with Amir starting. Each move consists of a player choosing a card with the number $k$ such that $k < 1969$ whose top face is white, and then this player turns all cards at positions $k,k+1,\ldots,k+40.$ The last player who can make a legal move wins.
(a) Does the game necessarily end?
(b) Does there exist a winning strategy for the starting player?
[i]Also compare shortlist 2009, combinatorics problem C1.[/i]
2001 Czech-Polish-Slovak Match, 6
Points with integer coordinates in cartesian space are called lattice points. We color $2000$ lattice points blue and $2000$ other lattice points red in such a way that no two blue-red segments have a common interior point (a segment is blue-red if its two endpoints are colored blue and red). Consider the smallest rectangular parallelepiped that covers all the colored points.
(a) Prove that this rectangular parallelepiped covers at least $500,000$ lattice points.
(b) Give an example of a coloring for which the considered rectangular paralellepiped covers at most $8,000,000$ lattice points.
2017 Bundeswettbewerb Mathematik, 2
In a convex regular $35$-gon $15$ vertices are colored in red. Are there always three red vertices that make an isosceles triangle?
1993 Cono Sur Olympiad, 1
On a chess board ($8*8$) there are written the numbers $1$ to $64$: on the first line, from left to right, there are the numbers $1, 2, 3, ... , 8$; on the second line, from left to right, there are the numbers $9, 10, 11, ... , 16$;etc. The $\"+\"$ and $\"-\"$ signs are put to each number such that, in each line and in each column, there are $4$ $\"+\"$ signs and $4$ $\"-\"$ signs. Then, the $64$ numbers are added. Find all the possible values of this sum.
2010 Greece Team Selection Test, 2
In a blackboard there are $K$ circles in a row such that one of the numbers $1,...,K$ is assigned to each circle from the left to the right.
Change of situation of a circle is to write in it or erase the number which is assigned to it.At the beginning no number is written in its own circle.
For every positive divisor $d$ of $K$ ,$1\leq d\leq K$ we change the situation of the circles in which their assigned numbers are divisible by $d$,performing for each divisor $d$ $K$ changes of situation.
Determine the value of $K$ for which the following holds;when this procedure is applied once for all positive divisors of $K$ ,then all numbers $1,2,3,...,K$ are written in the circles they were assigned in.
2014 Iran Team Selection Test, 3
we named a $n*n$ table $selfish$ if we number the row and column with $0,1,2,3,...,n-1$.(from left to right an from up to down)
for every {$ i,j\in{0,1,2,...,n-1}$} the number of cell $(i,j)$ is equal to the number of number $i$ in the row $j$.
for example we have such table for $n=5$
1 0 3 3 4
1 3 2 1 1
0 1 0 1 0
2 1 0 0 0
1 0 0 0 0
prove that for $n>5$ there is no $selfish$ table
2005 MOP Homework, 5
A group consists of $n$ tourists. Among every three of them there are at least two that are not familiar. For any partition of the group into two busses, there are at least two familiar tourists in one bus. Prove that there is a tourist who is familiar with at most two fifth of all the tourists.
2013 South East Mathematical Olympiad, 7
Given a $3\times 3$ grid, we call the remainder of the grid an “[i]angle[/i]” when a $2\times 2$ grid is cut out from the grid. Now we place some [i]angles[/i] on a $10\times 10$ grid such that the borders of those [i]angles[/i] must lie on the grid lines or its borders, moreover there is no overlap among the [i]angles[/i]. Determine the maximal value of $k$, such that no matter how we place $k$ [i]angles[/i] on the grid, we can always place another [i]angle[/i] on the grid.
2013 Iran MO (3rd Round), 6
Planet Tarator is a planet in the Yoghurty way galaxy. This planet has a shape of convex $1392$-hedron. On earth we don't have any other information about sides of planet tarator.
We have discovered that each side of the planet is a country, and has it's own currency. Each two neighbour countries have their own constant exchange rate, regardless of other exchange rates. Anybody who travels on land and crosses the border must change all his money to the currency of the destination country, and there's no other way to change the money. Incredibly, a person's money may change after crossing some borders and getting back to the point he started, but it's guaranteed that crossing a border and then coming back doesn't change the money.
On a research project a group of tourists were chosen and given same amount of money to travel around the Tarator planet and come back to the point they started. They always travel on land and their path is a nonplanar polygon which doesn't intersect itself. What is the maximum number of tourists that may have a pairwise different final amount of money?
[b]Note 1:[/b] Tourists spend no money during travel!
[b]Note 2:[/b] The only constant of the problem is 1392, the number of the sides. The exchange rates and the way the sides are arranged are unknown. Answer must be a constant number, regardless of the variables.
[b]Note 3:[/b] The maximum must be among all possible polyhedras.
Time allowed for this problem was 90 minutes.
2011 District Round (Round II), 4
Let $M$ be a set of six distinct positive integers whose sum is $60$. These numbers are written on the faces of a cube, one number to each face. A [i]move[/i] consists of choosing three faces of the cube that share a common vertex and adding $1$ to the numbers on those faces. Determine the number of sets $M$ for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.
2004 Postal Coaching, 5
How many paths from $(0,0)$ to $(n,n)$ of length $2n$ are there with exactly $k$ steps. A step is an occurence of the pair $EN$ in the path
1994 China Team Selection Test, 3
For any 2 convex polygons $S$ and $T$, if all the vertices of $S$ are vertices of $T$, call $S$ a sub-polygon of $T$.
[b]I. [/b]Prove that for an odd number $n \geq 5$, there exists $m$ sub-polygons of a convex $n$-gon such that they do not share any edges, and every edge and diagonal of the $n$-gon are edges of the $m$ sub-polygons.
[b]II.[/b] Find the smallest possible value of $m$.
2008 Ukraine Team Selection Test, 7
There is graph $ G_0$ on vertices $ A_1, A_2, \ldots, A_n$. Graph $ G_{n \plus{} 1}$ on vertices $ A_1, A_2, \ldots, A_n$ is constructed by the rule: $ A_i$ and $ A_j$ are joined only if in graph $ G_n$ there is a vertices $ A_k\neq A_i, A_j$ such that $ A_k$ is joined with both $ A_i$ and $ A_j$. Prove that the sequence $ \{G_n\}_{n\in\mathbb{N}}$ is periodic after some term with period $ T \le 2^n$.
2009 Mexico National Olympiad, 1
Let $n>1$ be an odd integer, and let $a_1$, $a_2$, $\dots$, $a_n$ be distinct real numbers. Let $M$ be the maximum of these numbers and $m$ the minimum. Show that it is possible to choose the signs of the expression $s=\pm a_1\pm a_2\pm\dots\pm a_n$ so that
\[m<s<M\]
2011 Estonia Team Selection Test, 6
On a square board with $m$ rows and $n$ columns, where $m\le n$, some squares are colored black in such a way that no two rows are alike. Find tha biggest integer $k$ such that, for every possible coloring to start with, one can always color $k$ columns entirely red in such a way that still no two rows are alike.