Found problems: 1488
2007 Tournament Of Towns, 7
There are $100$ boxes, each containing either a red cube or a blue cube. Alex has a sum of money initially, and places bets on the colour of the cube in each box in turn. The bet can be anywhere from $0$ up to everything he has at the time. After the bet has been placed, the box is opened. If Alex loses, his bet will be taken away. If he wins, he will get his bet back, plus a sum equal to the bet. Then he moves onto the next box, until he has bet on the last one, or until he runs out of money. What is the maximum factor by which he can guarantee to increase his amount of money, if he knows that the exact number of blue cubes is
[list][b](a)[/b] $1$;
[b](b)[/b] some integer $k$, $1 < k \leq 100$.[/list]
2014 Spain Mathematical Olympiad, 3
$60$ points are on the interior of a unit circle (a circle with radius $1$). Show that there exists a point $V$ on the circumference of the circle such that the sum of the distances from $V$ to the $60$ points is less than or equal to $80$.
1996 Baltic Way, 20
Is it possible to partition all positive integers into disjoint sets $A$ and $B$ such that
(i) no three numbers of $A$ form an arithmetic progression,
(ii) no infinite non-constant arithmetic progression can be formed by numbers of $B$?
2009 South africa National Olympiad, 3
Ten girls, numbered from 1 to 10, sit at a round table, in a random order. Each girl then receives a new number, namely the sum of her own number and those of her two neighbours. Prove that some girl receives a new number greater than 17.
2006 QEDMO 2nd, 2
There are $N$ cities in the country. Any two of them are connected either by a road or by an airway. A tourist wants to visit every city exactly once and return to the city at which he started the trip. Prove that he can choose a starting city and make a path, changing means of transportation at most once.
2014 Contests, 3
Given a regular 103-sided polygon. 79 vertices are colored red and the remaining vertices are colored blue. Let $A$ be the number of pairs of adjacent red vertices and $B$ be the number of pairs of adjacent blue vertices.
a) Find all possible values of pair $(A,B).$
b) Determine the number of pairwise non-similar colorings of the polygon satisfying $B=14.$ 2 colorings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon.
2001 All-Russian Olympiad, 1
The total mass of $100$ given weights with positive masses equals $2S$. A natural number $k$ is called [i]middle[/i] if some $k$ of the given weights have the total mass $S$. Find the maximum possible number of middle numbers.
1992 China Team Selection Test, 2
A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.
2005 All-Russian Olympiad, 4
A white plane is partitioned onto cells (in a usual way). A finite number of cells are coloured black. Each black cell has an even (0, 2 or 4) adjacent (by the side) white cells. Prove that one may colour each white cell in green or red such that every black cell will have equal number of red and green adjacent cells.
2003 Romania Team Selection Test, 12
A word is a sequence of n letters of the alphabet {a, b, c, d}. A word is said to be complicated if it contains two consecutive groups of identic letters. The words caab, baba and cababdc, for example, are complicated words, while bacba and dcbdc are not. A word that is not complicated is a simple word. Prove that the numbers of simple words with n letters is greater than $2^n$, if n is a positive integer.
2013 Romania Team Selection Test, 3
Given a positive integer $n$, consider a triangular array with entries $a_{ij}$ where $i$ ranges from $1$ to $n$ and $j$ ranges from $1$ to $n-i+1$. The entries of the array are all either $0$ or $1$, and, for all $i > 1$ and any associated $j$ , $a_{ij}$ is $0$ if $a_{i-1,j} = a_{i-1,j+1}$, and $a_{ij}$ is $1$ otherwise. Let $S$ denote the set of binary sequences of length $n$, and define a map $f \colon S \to S$ via $f \colon (a_{11}, a_{12},\cdots ,a_{1n}) \to (a_{n1}, a_{n-1,2}, \cdots , a_{1n})$. Determine the number of fixed points of $f$.
2003 Greece National Olympiad, 4
On the set $\Sigma$ of points of the plane $\Pi$ we define the operation $*$ which maps each pair $(X, Y )$ of points in $\Sigma$ to the point $Z = X * Y$ that is symmetric to $X$ with respect to $Y .$ Consider a square $ABCD$ in $\Pi$. Is it possible, using the points $A, B, C$ and applying the operation $*$ finitely many times, to construct the point $D?$
1989 IMO Longlists, 21
Let $ ABC$ be an equilateral triangle with side length equal to $ N \in \mathbb{N}.$ Consider the set $ S$ of all points $ M$ inside the triangle $ ABC$ satisfying
\[ \overrightarrow{AM} \equal{} \frac{1}{N} \cdot \left(n \cdot \overrightarrow{AB} \plus{} m \cdot \overrightarrow{AC} \right)\]
with $ m, n$ integers, $ 0 \leq n \leq N,$ $ 0 \leq m \leq N$ and $ n \plus{} m \leq N.$
Every point of S is colored in one of the three colors blue, white, red such that
[b](i) [/b]no point of $ S \cap [AB]$ is coloured blue
[b](ii)[/b] no point of $ S \cap [AC]$ is coloured white
[b](iii)[/b] no point of $ S \cap [BC]$ is coloured red
Prove that there exists an equilateral triangle the following properties:
[b](1)[/b] the three vertices of the triangle are points of $ S$ and coloured blue, white and red, respectively.
[b](2)[/b] the length of the sides of the triangle is equal to 1.
[i]Variant:[/i] Same problem but with a regular tetrahedron and four different colors used.
2001 Greece National Olympiad, 4
The numbers $1$ to $500$ are written on a board. Two pupils $A$ and $B$ play the following game: A player in turn deletes one of the numbers from the board. The game is over when only two numbers remain. Player $B$ wins if the sum of the two remaining numbers is divisible by $3,$ otherwise $A$ wins. If $A$ plays first, show that $B$ has a winning strategy.
2009 Serbia Team Selection Test, 3
Find the largest natural number $ n$ for which there exist different sets $ S_1,S_2,\ldots,S_n$ such that:
$ 1^\circ$ $ |S_i\cup S_j|\leq 2004$ for each two $ 1\leq i,j\le n$ and
$ 2^\circ$ $ S_i\cup S_j\cup S_k\equal{}\{1,2,\ldots,2008\}$ for each three integers $ 1\le i<j<k\le n$.
2002 All-Russian Olympiad, 4
On a plane are given finitely many red and blue lines, no two parallel, such that any intersection point of two lines of the same color also lies on another line of the other color. Prove that all the lines pass through a single point.
2003 Tournament Of Towns, 3
Can one cover a cube by three paper triangles (without overlapping)?
2003 Kurschak Competition, 2
Prove that if a graph $\mathcal{G}$ on $n\ge 3$ vertices has a unique $3$-coloring, then $\mathcal{G}$ has at least $2n-3$ edges.
(A graph is $3$-colorable when there exists a coloring of its vertices with $3$ colors such that no two vertices of the same color are connected by an edge. The graph can be $3$-colored uniquely if there do not exist vertices $u$ and $v$ of the graph that are painted different colors in one $3$-coloring, yet are colored the same in another.)
2002 All-Russian Olympiad, 4
There are some markets in a city. Some of them are joined by one-way streets, such that for any market there are exactly two streets to leave it. Prove that the city may be partitioned into $1014$ districts such that streets join only markets from different districts, and by the same one-way for any two districts (either only from first to second, or vice-versa).
2004 239 Open Mathematical Olympiad, 7
$200n$ diagonals are drawn in a convex $n$-gon. Prove that one of them intersects at least 10000 others.
[b]proposed by D. Karpov, S. Berlov[/b]
1999 China Team Selection Test, 3
For every permutation $ \tau$ of $ 1, 2, \ldots, 10$, $ \tau \equal{} (x_1, x_2, \ldots, x_{10})$, define $ S(\tau) \equal{} \sum_{k \equal{} 1}^{10} |2x_k \minus{} 3x_{k \minus{} 1}|$. Let $ x_{11} \equal{} x_1$. Find
[b]I.[/b] The maximum and minimum values of $ S(\tau)$.
[b]II.[/b] The number of $ \tau$ which lets $ S(\tau)$ attain its maximum.
[b]III.[/b] The number of $ \tau$ which lets $ S(\tau)$ attain its minimum.
2004 China Team Selection Test, 2
Let $ k$ be a positive integer. Set $ A \subseteq \mathbb{Z}$ is called a $ \textbf{k \minus{} set}$ if there exists $ x_1, x_2, \cdots, x_k \in \mathbb{Z}$ such that for any $ i \neq j$, $ (x_i \plus{} A) \cap (x_j \plus{} A) \equal{} \emptyset$, where $ x \plus{} A \equal{} \{ x \plus{} a \mid a \in A \}$. Prove that if $ A_i$ is $ \textbf{k}_i\textbf{ \minus{} set}$($ i \equal{} 1,2, \cdots, t$), and $ A_1 \cup A_2 \cup \cdots \cup A_t \equal{} \mathbb{Z}$, then $ \displaystyle \frac {1}{k_1} \plus{} \frac {1}{k_2} \plus{} \cdots \plus{} \frac {1}{k_t} \geq 1$.
2002 Finnish National High School Mathematics Competition, 3
$n$ pairs are formed from $n$ girls and $n$ boys at random.
What is the probability of having at least one pair of girls? For which $n$ the probability is over $0,9?$
2009 Germany Team Selection Test, 3
The 16 fields of a $4 \times 4$ checker board can be arranged in 18 lines as follows: the four lines, the four columns, the five diagonals from north west to south east and the five diagonals from north east to south west. These diagonals consists of 2,3 or 4 edge-adjacent fields of same colour; the corner fields of the chess board alone do not form a diagonal. Now, we put a token in 10 of the 16 fields. Each of the 18 lines contains an even number of tokens contains a point. What is the highest possible point number when can be achieved by optimal placing of the 10 tokens. Explain your answer.
2009 Germany Team Selection Test, 2
In Skinien there 2009 towns where each of them is connected with exactly 1004 other town by a highway. Prove that starting in an arbitrary town one can make a round trip along the highways such that each town is passed exactly once and finally one returns to its starting point.