Olympiad problems

1994 All-Russian Olympiad, 4

In a regular $ 6n\plus{}1$-gon, $ k$ vertices are painted in red and the others in blue. Prove that the number of isosceles triangles whose vertices are of the same color does not depend on the arrangement of the red vertices.

2002 Poland - Second Round, 3

Tags: search , geometry , inequalities , combinatorics unsolved , combinatorics

A positive integer $ n$ is given. In an association consisting of $ n$ members work $ 6$ commissions. Each commission contains at least $ \large \frac{n}{4}$ persons. Prove that there exist two commissions containing at least $ \large \frac{n}{30}$ persons in common.

1998 Brazil Team Selection Test, Problem 1

Tags: modular arithmetic , combinatorics unsolved , combinatorics

Let N be a positive integer greater than 2. We number the vertices of a regular 2n-gon clockwise with the numbers 1, 2, . . . ,N,−N,−N + 1, . . . ,−2,−1. Then we proceed to mark the vertices in the following way. In the first step we mark the vertex 1. If ni is the vertex marked in the i-th step, in the i+1-th step we mark the vertex that is |ni| vertices away from vertex ni, counting clockwise if ni is positive and counter-clockwise if ni is negative. This procedure is repeated till we reach a vertex that has already been marked. Let $f(N)$ be the number of non-marked vertices. (a) If $f(N) = 0$, prove that 2N + 1 is a prime number. (b) Compute $f(1997)$.

2017-IMOC, C1

Tags: combinatorics , combinatorics proposed , combinatorics unsolved

On a blackboard , the 2016 numbers $\frac{1}{2016} , \frac{2}{2016} ,... \frac{2016}{2016}$ are written. One can perfurm the following operation : Choose any numbers in the blackboard, say $a$ and$ b$ and replace them by $2ab-a-b+1$. After doing 2015 operation , there will only be one number $t$ Onthe blackboard . Find all possible values of $ t$.

2006 Junior Balkan MO, 4

Tags: geometry , rectangle , symmetry , floor function , combinatorics unsolved , combinatorics

Consider a $2n \times 2n$ board. From the $i$th line we remove the central $2(i-1)$ unit squares. What is the maximal number of rectangles $2 \times 1$ and $1 \times 2$ that can be placed on the obtained figure without overlapping or getting outside the board?

1997 IberoAmerican, 3

Tags: analytic geometry , combinatorics unsolved , combinatorics

Let $n \geq2$ be an integer number and $D_n$ the set of all the points $(x,y)$ in the plane such that its coordinates are integer numbers with: $-n \le x \le n$ and $-n \le y \le n$. (a) There are three possible colors in which the points of $D_n$ are painted with (each point has a unique color). Show that with any distribution of the colors, there are always two points of $D_n$ with the same color such that the line that contains them does not go through any other point of $D_n$. (b) Find a way to paint the points of $D_n$ with 4 colors such that if a line contains exactly two points of $D_n$, then, this points have different colors.

2003 Tournament Of Towns, 5

Tags: combinatorics unsolved , combinatorics

$25$ checkers are placed on $25$ leftmost squares of $1 \times N$ board. Checker can either move to the empty adjacent square to its right or jump over adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find minimal $N$ such that all the checkers could be placed in the row of $25$ successive squares but in the reverse order.

2011 Uzbekistan National Olympiad, 4

Tags: algebra , polynomial , geometry , geometric transformation , reflection , combinatorics unsolved , combinatorics

$A$ graph $G$ arises from $G_{1}$ and $G_{2}$ by pasting them along $S$ if $G$ has induced subgraphs $G_{1}$, $G_{2}$ with $G=G_{1}\cup G_{2}$ and $S$ is such that $S=G_{1}\cap G_{2}.$ A is graph is called [i]chordal[/i] if it can be constructed recursively by pasting along complete subgraphs, starting from complete subgraphs. For a graph $G(V,E)$ define its Hilbert polynomial $H_{G}(x)$ to be $H_{G}(x)=1+Vx+Ex^2+c(K_{3})x^3+c(K_{4})x^4+\ldots+c(K_{w(G)})x^{w(G)},$ where $c(K_{i})$ is the number of $i$-cliques in $G$ and $w(G)$ is the clique number of $G$. Prove that $H_{G}(-1)=0$ if and only if $G$ is chordal or a tree.

2003 China Western Mathematical Olympiad, 4

Tags: inequalities , function , vector , graph theory , combinatorics unsolved , combinatorics

$ 1650$ students are arranged in $ 22$ rows and $ 75$ columns. It is known that in any two columns, the number of pairs of students in the same row and of the same sex is not greater than $ 11$. Prove that the number of boys is not greater than $ 928$.

1994 All-Russian Olympiad Regional Round, 10.4

Tags: geometry , rectangle , combinatorics unsolved , combinatorics

A rectangle of size $ m \times n$ has been filled completely by trominoes (a tromino is an L-shape consisting of 3 unit squares). There are four ways to place a tromino 1st way: let the "corner" of the L be on top left 2nd way: let the "corner" of the L be on top right 3rd way: let the "corner" of the L be on bottom left 4th way: let the "corner" of the L be on bottom right Prove that the difference between the number of trominoes placed in the 1st and the 4th way is divisible by $ 3$.

1990 Austrian-Polish Competition, 5

Tags: function , combinatorics unsolved , combinatorics

Let $n>1$ be an integer and let $f_1$, $f_2$, ..., $f_{n!}$ be the $n!$ permutations of $1$, $2$, ..., $n$. (Each $f_i$ is a bijective function from $\{1,2,...,n\}$ to itself.) For each permutation $f_i$, let us define $S(f_i)=\sum^n_{k=1} |f_i(k)-k|$. Find $\frac{1}{n!} \sum^{n!}_{i=1} S(f_i)$.

2022 Grosman Mathematical Olympiad, P7

Tags: combinatorics , combinatorics unsolved

Let $k\leq n$ be two positive integers. $n$ points are marked on a line. It is known that for each marked point, the number of marked points at a distance $\leq 1$ from it (including the point itself) is divisible by $k$. Show that $k$ divides $n$ (without remainder).

2009 Indonesia MO, 1

Tags: probability , LaTeX , complementary counting , special factorizations , combinatorics unsolved , combinatorics

In a drawer, there are at most $ 2009$ balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is $ \frac12$. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?

1993 China National Olympiad, 5

Tags: combinatorics unsolved , combinatorics , easy

$10$ students bought some books in a bookstore. It is known that every student bought exactly three kinds of books, and any two of them shared at least one kind of book. Determine, with proof, how many students bought the most popular book at least? (Note: the most popular book means most students bought this kind of book)

1995 China Team Selection Test, 1

Tags: combinatorics unsolved , combinatorics

Let $S = \lbrace A = (a_1, \ldots, a_s) \mid a_i = 0$ or $1, i = 1, \ldots, 8 \rbrace$. For any 2 elements of $S$, $A = \lbrace a_1, \ldots, a_8\rbrace$ and $B = \lbrace b_1, \ldots, b_8\rbrace$. Let $d(A,B) = \sum_{i=1}{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most how many elements can $S$ have such that the distance between any 2 sets is at least 5?

2006 USA Team Selection Test, 5

Tags: induction , function , combinatorics unsolved , combinatorics

Let $n$ be a given integer with $n$ greater than $7$ , and let $\mathcal{P}$ be a convex polygon with $n$ sides. Any set of $n-3$ diagonals of $\mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $\mathcal{P}$ into $n-2$ triangles. A triangle in the triangulation of $\mathcal{P}$ is an interior triangle if all of its sides are diagonals of $\mathcal{P}$. Express, in terms of $n$, the number of triangulations of $\mathcal{P}$ with exactly two interior triangles, in closed form.

2009 Tuymaada Olympiad, 2

Tags: ceiling function , ratio , combinatorics unsolved , combinatorics

A necklace consists of 100 blue and several red beads. It is known that every segment of the necklace containing 8 blue beads contain also at least 5 red beads. What minimum number of red beads can be in the necklace? [i]Proposed by A. Golovanov[/i]

2014 Iran Team Selection Test, 6

Tags: combinatorics unsolved , combinatorics

Consider $n$ segments in the plane which no two intersect and between their $2n$ endpoints no three are collinear. Is the following statement true? Statement: There exists a simple $2n$-gon such that it's vertices are the $2n$ endpoints of the segments and each segment is either completely inside the polygon or an edge of the polygon.

2001 Tournament Of Towns, 4

Tags: combinatorics unsolved , combinatorics

Two persons play a game on a board divided into $3\times 100$ squares. They move in turn: the first places tiles of size $1\times2$ lengthwise (along the long axis of the board), the second, in the perpendicular direction. The loser is the one who cannot make a move. Which of the players can always win (no matter how his opponent plays), and what is the winning strategy?

2011 Postal Coaching, 5

Tags: geometry , rectangle , inequalities , combinatorics unsolved , combinatorics

The seats in the Parliament of some country are arranged in a rectangle of $10$ rows of $10$ seats each. All the $100$ $MP$s have diﬀerent salaries. Each of them asks all his neighbours (sitting next to, in front of, or behind him, i.e. $4$ members at most) how much they earn. They feel a lot of envy towards each other: an $MP$ is content with his salary only if he has at most one neighbour who earns more than himself. What is the maximum possible number of $MP$s who are satisﬁed with their salaries?

2014 Tuymaada Olympiad, 5

Tags: combinatorics unsolved , combinatorics , Tuymaada

There is an even number of cards on a table; a positive integer is written on each card. Let $a_k$ be the number of cards having $k$ written on them. It is known that \[a_n-a_{n-1}+a_{n-2}- \cdots \ge 0 \] for each positive integer $n$. Prove that the cards can be partitioned into pairs so that the numbers in each pair differ by $1$. [i](A. Golovanov)[/i]

2004 Tournament Of Towns, 2

Tags: combinatorics unsolved , combinatorics

Two persons are playing the following game. They have a pile of stones and take turns removing stones from it, with the first player taking the first turn. At each turn, the first player removes either 1 or 10 stones from the pile, and the second player removes either m or n stones. The player who can not make his move loses. It is known that for any number of stones in the pile, the first player can always win (regardless of the second player's moves). What are the possible values of m and n?

1998 South africa National Olympiad, 6

Tags: geometry , combinatorics unsolved , combinatorics

You are given $n$ squares, not necessarily all of the same size, which have total area 1. Is it always possible to place them without overlapping in a square of area 2?

2014 Iran MO (3rd Round), 5

Tags: combinatorics unsolved , combinatorics

A not necessary nonplanar polygon in $\mathbb{R}^3$ is called [b]Grid Polygon[/b] if each of it's edges are parallel to one of the axes. (a) There's a right angle between each two neighbour sides of the grid polygon, the plane containing this angle could be parallel to either $xy$ plane, $yz$ plane, or $xz$ plane. Prove that parity of the number of the angles that the plane containing each of them is parallel to $xy$ plane is equal to parity of the number of the angles that the plane containing each of them is parallel to $yz$ plane and parity of the number of the angles that the plane containing each of them is parallel to $zx$ plane. (b) A grid polygon is called [b]Inscribed[/b] if there's a point in the space that has an equal distance from all of the vertices of the polygon. Prove that any nonplanar grid hexagon is inscribed. (c) Does there exist a grid 2014-gon without repeated vertices such that there exists a plane that intersects all of it's edges? (d) If $a,b,c \in \mathbb{N}-\{1\}$, prove that $a,b,c$ are sidelengths of a triangle iff there exists a grid polygon in which the number of it's edges that are parallel to $x$ axis is $a$, the number of it's edges that are parallel to $y$ axis is $b$ and the number of it's edges that are parallel to $z$ axis is $c$. Time allowed for this exam was 1 hour.

2010 Korea - Final Round, 5

Tags: induction , symmetry , combinatorics unsolved , combinatorics

On a circular table are sitting $ 2n$ people, equally spaced in between. $ m$ cookies are given to these people, and they give cookies to their neighbors according to the following rule. (i) One may give cookies only to people adjacent to himself. (ii) In order to give a cookie to one's neighbor, one must eat a cookie. Select arbitrarily a person $ A$ sitting on the table. Find the minimum value $ m$ such that there is a strategy in which $ A$ can eventually receive a cookie, independent of the distribution of cookies at the beginning.