Olympiad problems

2010 Germany Team Selection Test, 2

Tags: geometry , rectangle , combinatorics , dissection , IMO Shortlist , Chessboard , perimeter

For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

1963 Swedish Mathematical Competition., 2

Tags: geometry , Chessboard , Squares , max

The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?

2015 Mathematical Talent Reward Programme, SAQ: P 3

Tags: combinatorics , Chessboard

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)

2008 BAMO, 2

Tags: Coloring , Chessboard , combinatorics

Consider a $7\times7$ chessboard that starts out with all the squares white. We start painting squares black, one at a time, according to the rule that after painting the first square, each newly painted square must be adjacent along a side to only the square just previously painted. The final figure painted will be a connected “snake” of squares. (a) Show that it is possible to paint $31$ squares. (b) Show that it is possible to paint $32$ squares. (c) Show that it is possible to paint $33$ squares.

2020 Romania EGMO TST, P2

Tags: combinatorics , romania , Chessboard

Let $n$ be a positive integer. In how many ways can we mark cells on an $n\times n$ board such that no two rows and no two columns have the same number of marked cells? [i]Selim Bahadir, Turkey[/i]

1999 Tournament Of Towns, 6

Tags: combinatorics , Chessboard , Rook

A rook is allowed to move one cell either horizontally or vertically. After $64$ moves the rook visited all cells of the $8 \times 8$ chessboard and returned back to the initial cell. Prove that the number of moves in the vertical direction and the number of moves in the horizontal direction cannot be equal. (A Shapovalov, R Sadykov)

2010 Lithuania National Olympiad, 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.

2022 Olympic Revenge, Problem 3

Tags: combinatorics , Chessboard

positive real $C$ is $n-vengeful$ if it is possible to color the cells of an $n \times n$ chessboard such that: i) There is an equal number of cells of each color. ii) In each row or column, at least $Cn$ cells have the same color. a) Show that $\frac{3}{4}$ is $n-vengeful$ for infinitely many values of $n$. b) Show that it does not exist $n$ such that $\frac{4}{5}$ is $n-vengeful$.

2015 Balkan MO Shortlist, C3

Tags: combinatorics , covering , dominoes , Chessboard

A chessboard $1000 \times 1000$ is covered by dominoes $1 \times 10$ that can be rotated. We don't know which is the cover, but we are looking for it. For this reason, we choose a few $N$ cells of the chessboard, for which we know the position of the dominoes that cover them. Which is the minimum $N$ such that after the choice of $N$ and knowing the dominoed that cover them, we can be sure and for the rest of the cover? (Bulgaria)

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)

2016 IFYM, Sozopol, 2

Tags: combinatorics , Chessboard

A cell is cut from a chessboard $8\, x\, 8$, after which an open broken line was built, which vertices are the centers of the remaining cells. Each segment of the broken line has a length $\sqrt{17}$ or $\sqrt{65}$. When is the number of such broken lines bigger – when the cut cell is $(1,2)$ or $(3,6)$? (The rows and columns on the board are numerated consecutively from 1 to 8.)

2014 Saudi Arabia Pre-TST, 1.4

Tags: combinatorics , Coloring , Chessboard

Majid wants to color the cells of an $n\times n$ chessboard into white and black so that each $2\times 2$ subsquare contains two white cells and two black cells. In how many ways can Majid color this $n\times n$ chessboard?