Olympiad problems

2011 NZMOC Camp Selection Problems, 1

Tags: combinatorics

A three by three square is filled with positive integers. Each row contains three different integers, the sums of each row are all the same, and the products of each row are all different. What is the smallest possible value for the sum of each row?

2018 Bulgaria JBMO TST, 4

Tags: inequalities , combinatorics

Each cell of an infinite table (infinite in all directions) is colored with one of $n$ given colors. All six cells of any $2\times 3$ (or $3 \times 2$) rectangle have different colors. Find the smallest possible value of $n$.

1991 IMO Shortlist, 9

Tags: combinatorics , point set , turan s theorem , extremal combinatorics , extremal graph theory

In the plane we are given a set $ E$ of 1991 points, and certain pairs of these points are joined with a path. We suppose that for every point of $ E,$ there exist at least 1593 other points of $ E$ to which it is joined by a path. Show that there exist six points of $ E$ every pair of which are joined by a path. [i]Alternative version:[/i] Is it possible to find a set $ E$ of 1991 points in the plane and paths joining certain pairs of the points in $ E$ such that every point of $ E$ is joined with a path to at least 1592 other points of $ E,$ and in every subset of six points of $ E$ there exist at least two points that are not joined?

2000 Tuymaada Olympiad, 7

Tags: combinatorial geometry , geometry , pentagon , regular polygon , combinatorics

Every two of five regular pentagons on the plane have a common point. Is it true that some of these pentagons have a common point?

2023 Iran MO (2nd Round), P4

Tags: combinatorics

4. A positive integer n is given.Find the smallest $k$ such that we can fill a $3*k$ gird with non-negative integers such that: $\newline$ $i$) Sum of the numbers in each column is $n$. $ii$) Each of the numbers $0,1,\dots,n$ appears at least once in each row.

1980 Austrian-Polish Competition, 4

Tags: counting , combinatorics , summation , induction

Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)

2014 Greece National Olympiad, 3

Tags: combinatorial geometry , modular arithmetic , euler , discrete intermediate value theorem , combinatorics

For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\frac{S_1}{S_2}=\frac{39}{64}.$

2013 IMO Shortlist, C5

Tags: additive combinatorics , sequence , function , combinatorics

Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

2003 Mexico National Olympiad, 6

Tags: combinatorics

Given a positive integer $n$, an allowed move is to form $2n+1$ or $3n+2$. The set $S_{n}$ is the set of all numbers that can be obtained by a sequence of allowed moves starting with $n$. For example, we can form $5 \rightarrow 11 \rightarrow 35$ so $5, 11$ and $35$ belong to $S_{5}$. We call $m$ and $n$ compatible if $S_{m}$ and $S_{n}$ has a common element. Which members of $\{1, 2, 3, ... , 2002\}$ are compatible with $2003$?

1970 IMO Longlists, 57

Tags: combinatorics

Let the numbers $1, 2, \ldots , n^2$ be written in the cells of an $n \times n$ square board so that the entries in each column are arranged increasingly. What are the smallest and greatest possible sums of the numbers in the $k^{th}$ row? ($k$ a positive integer, $1 \leq k \leq n$.)

2024 Assara - South Russian Girl's MO, 3

Tags: combinatorics , abstract algebra

In the cells of the $4\times N$ table, integers are written, modulo no more than $2024$ (i.e. numbers from the set $\{-2024, -2023,\dots , -2, -1, 0, 1, 2, 3,\dots , 2024\}$) so that in each of the four lines there are no two equal numbers. At what maximum $N$ could it turn out that in each column the sum of the numbers is equal to $2$? [i]G.M.Sharafetdinova[/i]

2002 Bundeswettbewerb Mathematik, 3

Tags: geometry , induction , combinatorics

Given a convex polyhedron with an even number of edges. Prove that we can attach an arrow to each edge, such that for every vertex of the polyhedron, the number of the arrows ending in this vertex is even.

2006 France Team Selection Test, 3

Tags: combinatorics , induction

Let $M=\{1,2,\ldots,3 \cdot n\}$. Partition $M$ into three sets $A,B,C$ which $card$ $A$ $=$ $card$ $B$ $=$ $card$ $C$ $=$ $n .$ Prove that there exists $a$ in $A,b$ in $B, c$ in $C$ such that or $a=b+c,$ or $b=c+a,$ or $c=a+b$ [i]Edited by orl.[/i]

2008 China Western Mathematical Olympiad, 3

Tags: combinatorics

For a given positive integer $n$, find the greatest positive integer $k$, such that there exist three sets of $k$ non-negative distinct integers, $A=\{x_1,x_2,\cdots,x_k\}, B=\{y_1,y_2,\cdots,y_k\}$ and $C=\{z_1,z_2,\cdots,z_k\}$ with $ x_j\plus{}y_j\plus{}z_j\equal{}n$ for any $ 1\leq j\leq k$. [size=85][color=#0000FF][Moderator edit: LaTeXified][/color][/size]

1998 Tournament Of Towns, 2

Tags: tiling , rectangle , combinatorial geometry , geometry , combinatorics

A square of side $1$ is divided into rectangles . We choose one of the two smaller sides of each rectangle (if the rectangle is a square, then we choose any of the four sides) . Prove that the sum of the lengths of all the chosen sides is at least $1$ . (Folklore)

2020 Thailand Mathematical Olympiad, 9

Tags: combinatorics

Let $n,k$ be positive integers such that $n>k$. There is a square-shaped plot of land, which is divided into $n\times n$ grid so that each cell has the same size. The land needs to be plowed by $k$ tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell sharing a common side until it reaches the upper-right corner cell. In addition, each tractor can only move in two directions: up and right. Determine the minimum possible number of unplowed cells.

1980 Bundeswettbewerb Mathematik, 3

Tags: combinatorics , combinatorial geometry

Given 2n+3 points in the plane, no three on a line and no four on a circle, prove that it is always possible to find a circle C that goes through three of the given points and splits the other 2n in half, that is, has n on the inside and n on the outside.

1972 Czech and Slovak Olympiad III A, 5

Tags: subset , combinatorics

Determine how many unordered pairs $\{A,B\}$ is there such that $A,B\subseteq\{1,\ldots,n\}$ and $A\cap B=\emptyset.$

II Soros Olympiad 1995 - 96 (Russia), 10.3

Tags: combinatorics

Points $A$, $B$, $C$, $D$ and $E$ are placed on the circle. In how many ways can the resulting five arcs be designated by the letters $a$, $b$, $c$, $d$ and $e$, if it is forbidden to designate an arc with the same letter as one of its ends? (For example, an arc with ends $A$ and $B$ cannot be designated by the letter $a$ or $b$.)

2010 BAMO, 4

Tags: chessboard , combinatorics

Place eight rooks on a standard $8 \times 8$ chessboard so that no two are in the same row or column. With the standard rules of chess, this means that no two rooks are attacking each other. Now paint $27$ of the remaining squares (not currently occupied by rooks) red. Prove that no matter how the rooks are arranged and which set of $27$ squares are painted, it is always possible to move some or all of the rooks so that: • All the rooks are still on unpainted squares. • The rooks are still not attacking each other (no two are in the same row or same column). • At least one formerly empty square now has a rook on it; that is, the rooks are not on the same $8$ squares as before.

1947 Moscow Mathematical Olympiad, 135

Tags: convex , combinatorial geometry , combinatorics , geometry

a) Given $5$ points on a plane, no three of which lie on one line. Prove that four of these points can be taken as vertices of a convex quadrilateral. b) Inside a square, consider a convex quadrilateral and inside the quadrilateral, take a point $A$. It so happens that no three of the $9$ points — the vertices of the square, of the quadrilateral and $A$ — lie on one line. Prove that $5$ of these points are vertices of a convex pentagon.

1993 Balkan MO, 2

Tags: calculus , integration , inequalities , floor function , combinatorics , ceiling function

A positive integer given in decimal representation $\overline{ a_na_{n-1} \ldots a_1a_0 }$ is called [i]monotone[/i] if $a_n\leq a_{n-1} \leq \cdots \leq a_0$. Determine the number of monotone positive integers with at most 1993 digits.

2010 Iran MO (3rd Round), 1

Tags: geometry , 3d geometry , symmetry , geometric transformation , combinatorics , rotation

Prove that the group of orientation-preserving symmetries of the cube is isomorphic to $S_4$ (the group of permutations of $\{1,2,3,4\}$).(20 points)

2015 Azerbaijan Team Selection Test, 2

Tags: combinatorics

Alex and Bob play a game 2015 x 2015 checkered board by the following rules.Initially the board is empty: the players move in turn, Alex moves first. By a move, a player puts either red or blue token into any unoccopied square. If after a player's move there appears a row of three consecutive tokens of the same color( this row may be vertical,horizontal, or dioganal), then this player wins. If all the cells are occupied by tokens, but no such row appears, then a draw is declared.Determine whether Alex, Bob, or none of them has winning strategy.

2015 Ukraine Team Selection Test, 9

Tags: point , heptagon , convex , combinatorial geometry , pentagon , combinatorics

The set $M$ consists of $n$ points on the plane and satisfies the conditions: $\bullet$ there are $7$ points in the set $M$, which are vertices of a convex heptagon, $\bullet$ for arbitrary five points with $M$, which are vertices of a convex pentagon, there is a point that also belongs to $M$ and lies inside this pentagon. Find the smallest possible value that $n$ can take .