Found problems: 1800
2005 Romania Team Selection Test, 2
On the edges of a convex polyhedra we draw arrows such that from each vertex at least an arrow is pointing in and at least one is pointing out.
Prove that there exists a face of the polyhedra such that the arrows on its edges form a circuit.
[i]Dan Schwartz[/i]
2002 Irish Math Olympiad, 2
$ (a)$ A group of people attends a party. Each person has at most three acquaintances in the group, and if two people do not know each other, then they have a common acquaintance in the group. What is the maximum possible number of people present?
$ (b)$ If, in addition, the group contains three mutual acquaintances, what is the maximum possible number of people?
2008 China Team Selection Test, 3
Let $ S$ be a set that contains $ n$ elements. Let $ A_{1},A_{2},\cdots,A_{k}$ be $ k$ distinct subsets of $ S$, where $ k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k)$. Prove that the number of subsets of $ S$ that don't contain any $ A_{i} (1\leq i\leq k)$ is greater than or equal to $ 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).$
1951 Miklós Schweitzer, 13
Of how many terms does the expansion of a determinant of order $ 2n$ consist if those and only those elements $ a_{ik}$ are non-zero for which $ i\minus{}k$ is divisible by $ n$?
1997 All-Russian Olympiad, 2
An $n\times n$ square grid ($n\geqslant 3$) is rolled into a cylinder. Some of the cells are then colored black. Show that there exist two parallel lines (horizontal, vertical or diagonal) of cells containing the same number of black cells.
[i]E. Poroshenko[/i]
2009 Hong kong National Olympiad, 2
there are $n$ points on the plane,any two vertex are connected by an edge of red,yellow or green,and any triangle with vertex in the graph contains exactly $2$ colours.prove that $n<13$
2005 Iran MO (3rd Round), 3
$f(n)$ is the least number that there exist a $f(n)-$mino that contains every $n-$mino.
Prove that $10000\leq f(1384)\leq960000$.
Find some bound for $f(n)$
2003 Baltic Way, 9
It is known that $n$ is a positive integer and $n \le 144$. Ten questions of the type “Is $n$ smaller than $a$?” are allowed. Answers are given with a delay: for $i = 1, \ldots , 9$, the $i$-th question is answered only after the $(i + 1)$-th question is asked. The answer to the tenth question is given immediately.
Find a strategy for identifying $n$.
2010 Slovenia National Olympiad, 5
Let $ABCD$ be a square with the side of $20$ units. Amir divides this square into $400$ unit squares. Reza then picks $4$ of the vertices of these unit squares. These vertices lie inside the square $ABCD$ and define a rectangle with the sides parallel to the sides of the square $ABCD.$ There are exactly $24$ unit squares which have at least one point in common with the sides of this rectangle. Find all possible values for the area of a rectangle with these properties.
[hide="Note"][i]Note:[/i] Vid changed to Amir, and Eva change to Reza![/hide]
2006 Bundeswettbewerb Mathematik, 1
A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors).
Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.)
[hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]
2014 Bundeswettbewerb Mathematik, 3
A line $g$ is given in a plane. $n$ distinct points are chosen arbitrarily from $g$ and are named as $A_1, A_2, \ldots, A_n$. For each pair of points $A_i,A_j$, a semicircle is drawn with $A_i$ and $A_j$ as its endpoints. All semicircles lie on the same side of $g$. Determine the maximum number of points (which are not lying in $g$) of intersection of semicircles as a function of $n$.
2001 Turkey MO (2nd round), 3
One wants to distribute $n$ same sized cakes between $k$ people equally by cutting every cake at most once. If the number of positive divisors of $n$ is denoted as $d(n)$, show that the number of different values of $k$ which makes such distribution possible is $n+d(n)$
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.
2013 Korea - Final Round, 6
For any permutation $ f : \{ 1, 2, \cdots , n \} \to \{1, 2, \cdots , n \} $, and define
\[ A = \{ i | i > f(i) \} \]
\[ B = \{ (i, j) | i<j \le f(j) < f(i) \ or \ f(j) < f(i) < i < j \} \]
\[ C = \{ (i, j) | i<j \le f(i) < f(j) \ or \ f(i) < f(j) < i < j \} \]
\[ D = \{ (i, j) | i< j \ and \ f(i) > f(j)\} \]
Prove that $ |A| + 2|B| + |C| = |D| $.
2001 Polish MO Finals, 3
Given positive integers $n_1<n_2<...<n_{2000}<10^{100}$. Prove that we can choose from the set $\{n_1,...,n_{2000}\}$ nonempty, disjont sets $A$ and $B$ which have the same number of elements, the same sum and the same sum of squares.
1971 IMO Longlists, 54
A set $M$ is formed of $\binom{2n}{n}$ men, $n=1,2,\ldots$. Prove that we can choose a subset $P$ of the set $M$ consisting of $n+1$ men such that one of the following conditions is satisfied:
$(1)$ every member of the set $P$ knows every other member of the set $P$;
$(2)$ no member of the set $P$ knows any other member of the set $P$.
2011 Baltic Way, 10
Two persons play the following game with integers. The initial number is $2011^{2011}$. The players move in turns. Each move consists of subtraction of an integer between $1$ and $2010$ inclusive, or division by $2011$, rounding down to the closest integer when necessary. The player who first obtains a non-positive integer wins. Which player has a winning strategy?
2008 Serbia National Math Olympiad, 4
Each point of a plane is painted in one of three colors. Show that there exists a triangle such that:
$ (i)$ all three vertices of the triangle are of the same color;
$ (ii)$ the radius of the circumcircle of the triangle is $ 2008$;
$ (iii)$ one angle of the triangle is either two or three times greater than one of the other two angles.
2005 Canada National Olympiad, 1
An equilateral triangle of side length $ n$ is divided into unit triangles. Let $ f(n)$ be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in a path share a common edge and the path never travels up (from a lower row to a higher row) or revisits a triangle. An example is shown on the picture for $ n \equal{} 5$. Determine the value of $ f(2005)$.
2013 India IMO Training Camp, 1
Let $n \ge 2$ be an integer. There are $n$ beads numbered $1, 2, \ldots, n$. Two necklaces made out of some of these beads are considered the same if we can get one by rotating the other (with no flipping allowed). For example, with $n \ge 5$, the necklace with four beads $1, 5, 3, 2$ in the clockwise order is same as the one with $5, 3, 2, 1$ in the clockwise order, but is different from the one with $1, 2, 3, 5$ in the clockwise order.
We denote by $D_0(n)$ (respectively $D_1(n)$) the number of ways in which we can use all the beads to make an even number (resp. an odd number) of necklaces each of length at least $3$. Prove that $n - 1$ divides $D_1(n) - D_0(n)$.
2007 QEDMO 4th, 3
Let $ n$ be a positive integer, and let $ M\equal{}\left\{ 1,2,...,n\right\}$. Two players take turns at the following game: Each player, at his turn, has to select an element of $ M$ and remove all divisors of this element (including this element itself) from the set $ M$.
[b]a)[/b] Assume that the player who cannot move anymore (because the set $ M$ is empty when it's his move) wins. For which values of $ n$ does the first player have a winning strategy?
[b]b)[/b] Assume that the player who cannot move anymore (because the set $ M$ is empty when it's his move) loses. For which values of $ n$ does the first player have a winning strategy?
2003 Italy TST, 2
For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A [i]tromino[/i] is an $L$-shape formed by three connected unit squares.
$(a)$ For which values of $n$ is it possible to cover all the black squares with non-overlapping trominoes lying entirely on the chessboard?
$(b)$ When it is possible, find the minimum number of trominoes needed.
2017 Baltic Way, 10
Maker and Breaker are building a wall. Maker has a supply of green cubical building blocks, and Breaker has a supply of red ones, all of the same size. On the ground, a row of $m$ squares has been marked in chalk as place-holders. Maker and Breaker now take turns in placing a block either directly on one of these squares, or on top of another block already in place, in such a way that the height of each column never exceeds $n$. Maker places the first block.
Maker bets that he can form a green row, i.e. all $m$ blocks at a certain height are green. Breaker bets that he can prevent Maker from achieving this. Determine all pairs $(m,n)$ of positive integers for which Maker can make sure he wins the bet.
2009 Italy TST, 1
Let $n,k$ be positive integers such that $n\ge k$. $n$ lamps are placed on a circle, which are all off. In any step we can change the state of $k$ consecutive lamps. In the following three cases, how many states of lamps are there in all $2^n$ possible states that can be obtained from the initial state by a certain series of operations?
i)$k$ is a prime number greater than $2$;
ii) $k$ is odd;
iii) $k$ is even.
1996 Turkey MO (2nd round), 3
Let $n$ integers on the real axis be colored. Determine for which positive integers $k$ there exists a family $K$ of closed intervals with the following properties:
i) The union of the intervals in $K$ contains all of the colored points;
ii) Any two distinct intervals in $K$ are disjoint;
iii) For each interval $I$ at $K$ we have ${{a}_{I}}=k.{{b}_{I}}$, where ${{a}_{I}}$ denotes the number of integers in $I$, and ${{b}_{I}}$ the number of colored integers in $I$.