Olympiad problems

2010 JBMO Shortlist, 2

Tags: geometry , rectangle , combinatorics , rotation , inequalities

A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.

2020 Canadian Junior Mathematical Olympiad, 5

Tags: combinatorics

There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exactly $2020$ different ways to select the coins in his purse and the sum of these selected coins is $2020$?

1976 All Soviet Union Mathematical Olympiad, 220

Tags: distance , combinatorics , combinatorial geometry

There are $50$ exact watches lying on a table. Prove that there exist a certain moment, when the sum of the distances from the centre of the table to the ends of the minute hands is more than the sum of the distances from the centre of the table to the centres of the watches.

1999 China Team Selection Test, 3

Tags: combinatorics

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.

2014 Cuba MO, 5

Tags: number theory , combinatorics

The number 2013 is written on a blackboard. Two players participate, alternating in turns, in the next game. A movement consists in changing the number that is on the board for the difference of this number and one of its divisors. The player who writes a zero loses. Which of the two players can guarantee victory?

1998 Bundeswettbewerb Mathematik, 2

Tags: subset , intersection , combinatorics

Prove that there exist $16$ subsets of set $M = \{1,2,...,10000\}$ with the following property: For every $z \in M$ there are eight of these subsets whose intersection is $\{z\}$.

2005 Junior Balkan Team Selection Tests - Romania, 3

Tags: combinatorics

In a country 6 cities are connected two by two with round-trip air routes operated by exactly one of the two air companies in that country. Prove that there exist 4 cities $A$, $B$, $C$ and $D$ such that each of the routes $A\leftrightarrow B$, $B\leftrightarrow C$, $C\leftrightarrow D$ and $D\leftrightarrow A$ are operated by the same company. [i]Dan Schwartz[/i]

2009 Brazil National Olympiad, 1

Tags: 3d geometry , geometry , combinatorics

Prove that there exists a positive integer $ n_0$ with the following property: for each integer $ n \geq n_0$ it is possible to partition a cube into $ n$ smaller cubes.

2020 South East Mathematical Olympiad, 8

Tags: probability , combinatorics , algebra

Using a nozzle to paint each square in a $1 \times n$ stripe, when the nozzle is aiming at the $i$-th square, the square is painted black, and simultaneously, its left and right neighboring square (if exists) each has an independent probability of $\tfrac{1}{2}$ to be painted black. In the optimal strategy (i.e. achieving least possible number of painting), the expectation of number of painting to paint all the squares black, is $T(n)$. Find the explicit formula of $T(n)$.

2019 Indonesia MO, 8

Tags: combinatorics

Let $n > 1$ be a positive integer and $a_1, a_2, \dots, a_{2n} \in \{ -n, -n + 1, \dots, n - 1, n \}$. Suppose \[ a_1 + a_2 + a_3 + \dots + a_{2n} = n + 1 \] Prove that some of $a_1, a_2, \dots, a_{2n}$ have sum 0.

2023 Silk Road, 2

Tags: coloring , domino , combinatorics

Let $n$ be a positive integer. Each cell of a $2n\times 2n$ square is painted in one of the $4n^2$ colors (with some colors may be missing). We will call any two-cell rectangle a [i]domino[/I], and a domino is called [i]colorful[/I] if its cells have different colors. Let $k$ be the total number of colorful dominoes in our square; $l$ be the maximum integer such that every partition of the square into dominoes contains at least $l$ colorful dominoes. Determine the maximum possible value of $4l-k$ over all possible colourings of the square.

2024 Saint Petersburg Mathematical Olympiad, 4

Tags: combinatorics

The coach lined up $200$ volleyball players and gave them $m$ balls (each volleyball player could get any number of balls). From time to time, one of the volleyball players throws the ball to another (and he catches it). After a while, it turned out that of any two volleyball players, the left one threw the ball to the right exactly twice, and the right one to the left exactly once. For which minimum $m$ is this possible?

1966 IMO Longlists, 45

Tags: extremal combinatorics , combinatorics of words , maximization , combinatorics

An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters.

2023 ISI Entrance UGB, 4

Tags: combinatorics , number theory

Let $n_1, n_2, \cdots , n_{51}$ be distinct natural numbers each of which has exactly $2023$ positive integer factors. For instance, $2^{2022}$ has exactly $2023$ positive integer factors $1,2, 2^{2}, 2^{3}, \cdots 2^{2021}, 2^{2022}$. Assume that no prime larger than $11$ divides any of the $n_{i}$'s. Show that there must be some perfect cube among the $n_{i}$'s.

1998 Croatia National Olympiad, Problem 3

Tags: fe , combinatorics , functional equation

Let $A=\{1,2,\ldots,2n\}$ and let the function $g:A\to A$ be defined by $g(k)=2n-k+1$. Does there exist a function $f:A\to A$ such that $f(k)\ne g(k)$ and $f(f(f(k)))=g(k)$ for all $k\in A$, if (a) $n=999$; (b) $n=1000$?

2024 Mongolian Mathematical Olympiad, 2

Tags: extremal , combinatorics , geometry

We call a triangle consisting of three vertices of a pentagon [i]big[/i] if it's area is larger than half of the pentagon's area. Find the maximum number of [i]big[/i] triangles that can be in a convex pentagon. [i]Proposed by Gonchigdorj Sandag[/i]

1998 Tournament Of Towns, 5

Tags: coloring , combinatorics

There are $20$ beads of $10$ colours, two of each colour. They are put in $10$ boxes. It is known that one bead can be selected from each of the boxes so that each colour is represented. Prove that the number of such selections is a non-zero power of $2$. (A Grishin)

KoMaL A Problems 2024/2025, A. 907

Tags: combinatorics

$2025$ light bulbs are operated by some switches. Each switch works on a subset of the light bulbs. When we use a switch, all the light bulbs in the subset change their state: bulbs that were on turn off, and bulbs that were off turn on. We know that every light bulb is operated by at least one of the switches. Initially, all lamps were off. Find the biggest number $k$ for which we can surely turn on at least $k$ light bulbs. [i]Based on an OKTV problem[/i]

2018 NZMOC Camp Selection Problems, 7

Tags: combinatorial geometry , coloring , combinatorics

Let $N$ be the number of ways to colour each cell in a $2 \times 50$ rectangle either red or blue such that each $2 \times 2$ block contains at least one blue cell. Show that $N$ is a multiple of $3^{25}$, but not a multiple of $3^{26}$

2018 JBMO TST-Turkey, 7

Tags: combinatorics

In the round robin chess tournament organized in a school every two students played one match among themselves. Find the minimal possible number of students in the school if each girl student has at least 21 wins in matches against boy students and each boy student has at least 12 wins in matches against girl students.

2020 Israel National Olympiad, 6

Tags: combinatorics

On a circle the numbers from 1 to 6 are written in order, as depicted in the picture. In each move, Lior picks a number $a$ on the circle whose neighbors are $b$ and $c$ and replaces it by the number $\frac{bc}{a}$. Can Lior reach a state in which the product of the numbers on the circle is greater than $10^{100}$ in [b]a)[/b] at most 100 moves [b]b)[/b] at most 110 moves

2009 Brazil National Olympiad, 3

Tags: floor function , chip-firing , search , analytic geometry , combinatorics

There are $ 2009$ pebbles in some points $ (x,y)$ with both coordinates integer. A operation consists in choosing a point $ (a,b)$ with four or more pebbles, removing four pebbles from $ (a,b)$ and putting one pebble in each of the points \[ (a,b\minus{}1),\ (a,b\plus{}1),\ (a\minus{}1,b),\ (a\plus{}1,b)\] Show that after a finite number of operations each point will necessarily have at most three pebbles. Prove that the final configuration doesn't depend on the order of the operations.

2022 BMT, Tie 3

Tags: combinatorics , number theory

Let $A$ be the product of all positive integers less than $1000$ whose ones or hundreds digit is $7$. Compute the remainder when $A/101$ is divided by $101$.

1952 Moscow Mathematical Olympiad, 230

Tags: combinatorics , game , game strategy

$200$ soldiers occupy in a rectangle (military call it a square and educated military a carree): $20$ men (per row) times $10$ men (per column). In each row, we consider the tallest man (if some are of equal height, choose any of them) and of the $10$ men considered we select the shortest (if some are of equal height, choose any of them). Call him $A$. Next the soldiers assume their initial positions and in each column the shortest soldier is selected, of these $20$, the tallest is chosen. Call him $B$. Two colonels bet on which of the two soldiers chosen by these two distinct procedures is taller: $A$ or $B$. Which colonel wins the bet?

1985 Tournament Of Towns, (089) 5

Tags: digit , square table , combinatorics

The digits $0, 1 , 2, ..., 9$ are written in a $10 x 10$ table , each number appearing $10$ times . (a) Is it possible to write them in such a way that in any row or column there would be not more than $4$ different digits? (b) Prove that there must be a row or column containing more than $3$ different digits . { L . D . Kurlyandchik , Leningrad)