Olympiad problems

2015 Mathematical Talent Reward Programme, SAQ: P 3

Tags: chessboard , combinatorics

Show that, in a chessboard, it is possible to traverse to any given square from another given square using a knight. (A knight can move in a chessboard by going two steps in one direction and one step in a perpendicular direction as shown in the given figure)

1986 Tournament Of Towns, (125) 7

Tags: chessboard , coloring , combinatorics , combinatorial geometry

Each square of a chessboard is painted either blue or red . Prove that the squares of one colour possess the property that the chess queen can perform a tour of all of them. The rules are that the queen may visit the squares of this colour not necessarily only once each , and may not be placed on squares of the other colour, although she may pass over them ; the queen moves along any horizontal , vertical or diagonal file over any distance. (A . K . Tolpugo , Kiev)

2006 ISI B.Math Entrance Exam, 1

Tags: chessboard , extremal combinatorics , combinatorics

Bishops on a chessboard move along the diagonals ( that is , on lines parallel to the two main diagonals ) . Prove that the maximum number of non-attacking bishops on an $n*n$ chessboard is $2n-2$. (Two bishops are said to be attacking if they are on a common diagonal).

2019 Tuymaada Olympiad, 3

Tags: minimum , minimum value , chessboard , combinatorics

The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to adjacent rooms. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess queen (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 2$)?

VMEO II 2005, 9

Tags: chessboard , queen , combinatorics

On a board with $64$ ($8 \times 8$) squares, find a way to arrange $9$ queens and $ 1$ king so that every queen cannot capture another queen.

2011 Belarus Team Selection Test, 3

Tags: chessboard , counting , combinatorics , matrix

2500 chess kings have to be placed on a $100 \times 100$ chessboard so that [b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); [b](ii)[/b] each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.) [i]Proposed by Sergei Berlov, Russia[/i]

2016 IFYM, Sozopol, 6

Tags: combinatorics , chessboard

We are given a chessboard 100 x 100, $k$ barriers (each with length 1), and one ball. We want to put the barriers between the cells of the board and put the ball in some cell, in such way that the ball can get to each possible cell on the board. The only way that the ball can move is by lifting the board so it can go only forward, backward, to the left or to the right. The ball passes all cells on its way until it reaches a barrier or the edge of the board where it stops. What’s the least number of barriers we need so we can achieve that?

2004 South East Mathematical Olympiad, 4

Tags: combinatorics , chessboard

Given a positive integer $n (n>2004)$, we put 1, 2, 3, …,$n^2$ into squares of an $n\times n$ chessboard with one number in a square. A square is called a “good square” if the square satisfies following conditions: 1) There are at least 2004 squares that are in the same row with the square such that any number within these 2004 squares is less than the number within the square. 2) There are at least 2004 squares that are in the same column with the square such that any number within these 2004 squares is less than the number within the square. Find the maximum value of the number of the “good square”.

1999 Tournament Of Towns, 6

Tags: rectangle , combinatorial geometry , chessboard , combinatorics

On a large chessboard $2n$ of its $1 \times 1$ squares have been marked such thar the rook (which moves only horizontally or vertically) can visit all the marked squares without jumpin over any unmarked ones. Prove that the figure consisting of all the marked squares can be cut into rectangles. (A Shapovalov)

2010 Contests, 3

Tags: combinatorics , chessboard

In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.

2010 Junior Balkan Team Selection Tests - Romania, 4

Tags: chessboard , combinatorics

An $8 \times 8$ chessboard consists of $64$ square units. In some of the unit squares of the board, diagonals are drawn so that any two diagonals have no common points. What is the maximum number of diagonals that can be drawn?

1978 Bundeswettbewerb Mathematik, 1

Tags: chessboard , knight , even , combinatorics

A knight is modified so that it moves $p$ fields horizontally or vertically and $q$ fields in the perpendicular direction. It is placed on an infinite chessboard. If the knight returns to the initial field after $n$ moves, show that $n$ must be even.

2020 Bundeswettbewerb Mathematik, 2

Tags: combinatorics , chess knight , chessboard

Konstantin moves a knight on a $n \times n$- chess board from the lower left corner to the lower right corner with the minimal number of moves. Then Isabelle takes the knight and moves it from the lower left corner to the upper right corner with the minimal number of moves. For which values of $n$ do they need the same number of moves?

2015 IFYM, Sozopol, 7

Tags: table , chessboard , coloring , combinatorics

A corner with arm $n$ is a figure made of $2n-1$ unit squares, such that 2 rectangles $1$ x $(n-1)$ are connected to two adjacent sides of a square $1$ x $1$, so that their unit sides coincide. The squares or a chessboard $100$ x $100$ are colored in 15 colors. We say that a corner with arm 8 is [i]“multicolored”[/i], if it contains each of the colors on the board. What’s the greatest number of corners with arm 8 which could be [i]“mutlticolored”[/i]?

2019 Saint Petersburg Mathematical Olympiad, 6

Tags: chessboard , infinite chessboard , positive integer , combinatorial geometry , combinatorics

Is it possible to arrange everything in all cells of an infinite checkered plane all natural numbers (once) so that for each $n$ in each square $n \times n$ the sum of the numbers is a multiple of $n$?

2011 QEDMO 8th, 1

Tags: combinatorial geometry , chessboard , combinatorics

A $T$-tetromino is a non-convex as well as non-rotationally symmetrical tetromino, which has a maximum number of outside corners (popularly also "Tetris Stone "called). Find all natural numbers $n$ for which, a $n \times n$ chessboard is found that can be covered only with such $T$-tetrominos.

2024 Tuymaada Olympiad, 5

Tags: chessboard , board , coloring , combinatorics

Given a board with size $25\times 25$. Some $1\times 1$ squares are marked, so that for each $13\times 13$ and $4\times 4$ sub-boards, there are atleast $\frac{1}{2}$ marked parts of the sub-board. Find the least possible amount of marked squares in the entire board.

2018 India PRMO, 26

Tags: chessboard , combinatorics

What is the number of ways in which one can choose $60$ unit squares from a $11 \times 11$ chessboard such that no two chosen squares have a side in common?

2005 Tournament of Towns, 4

Tags: chessboard

A chess piece moves as follows: it can jump 8 or 9 squares either vertically or horizontally. It is not allowed to visit the same square twice. At most, how many squares can this piece visit on a $15 \times 15$ board (it can start from any square)? [i](4 points)[/i]

2013 Balkan MO Shortlist, C5

Tags: chessboard , coloring , combinatorics

The cells of an $n \times n$ chessboard are coloured in several colours so that no $2\times 2$ square contains four cells of the same colour. A [i]proper path [/i] of length $m$ is a sequence $a_1,a_2,..., a_m$ of distinct cells in which the cells $a_i$ and $a_{i+1}$ have a common side and are coloured in different colours for all $1 \le i < m$. Show that there exists a proper path of length $n$.

2017 Saudi Arabia JBMO TST, 4

Tags: combinatorics , chessboard

Find the number of ways one can put numbers $1$ or $2$ in each cell of an $8\times 8$ chessboard in such a way that the sum of the numbers in each column and in each row is an odd number. (Two ways are considered different if the number in some cell in the first way is different from the number in the cell situated in the corresponding position in the second way)

1988 Tournament Of Towns, (198) 1

Tags: combinatorics , chessboard , combinatorial geometry

What is the smallest number of squares of a chess board that can be marked in such a manner that (a) no two marked squares may have a common side or a common vertex, and (b) any unmarked square has a common side or a common vertex with at least one marked square? Indicate a specific configuration of marked squares satisfying (a) and (b) and show that a lesser number of marked squares will not suffice. (A. Andjans, Riga)

2015 QEDMO 14th, 11

Tags: chessboard , combinatorial geometry , combinatorics

Let $m, n$ be natural numbers and let $m\cdot n$ be a multiple of $4$. A chessboard with $m \times n$ fields are covered with $1 \times 2$ large dominoes without gaps and without overlapping. Show that the number of dominoes that are parallel to a edge of the chess board is fixed . [hide=original wording] Seien m, n natu¨rliche Zahlen und sei m · n ein Vielfaches von 4. Ein Schachbrett mit m × n Feldern sei mit 1 × 2 großen Dominosteinen lu¨ckenlos und u¨berlappungsfrei u¨berdeckt. Zeige, dass die Anzahl der Dominosteine, die zu einer fest gew¨ahlten Kante des Schachbrettes parallel sind, gerade ist. [/hide]

2019 Saudi Arabia BMO TST, 3

Tags: chessboard , coloring , combinatorics

For $n \ge 3$, it is given an $2n \times 2n$ board with black and white squares. It is known that all border squares are black and no $2 \times 2$ subboard has all four squares of the same color. Prove that there exists a $2 \times 2$ subboard painted like a chessboard, i.e. with two opposite black corners and two opposite white corners.

2008 Bulgaria Team Selection Test, 1

Tags: graph theory , induction , hamiltonian path , chessboard , combinatorics

Let $n$ be a positive integer. There is a pawn in one of the cells of an $n\times n$ table. The pawn moves from an arbitrary cell of the $k$th column, $k \in \{1,2, \cdots, n \}$, to an arbitrary cell in the $k$th row. Prove that there exists a sequence of $n^{2}$ moves such that the pawn goes through every cell of the table and finishes in the starting cell.