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Omni-MATH / (a) Can 1000 queens be placed on a 2017×2017 chessboard such that every square is attacked by some queen? A square is attacked by a queen if it lies in the same row, column, or dia…

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(a) Can 1000 queens be placed on a 2017×20172017 \times 2017 chessboard such that every square is attacked by some queen? A square is attacked by a queen if it lies in the same row, column, or diagonal as the queen. (b) A 2017×20172017 \times 2017 grid of squares originally contains a 0 in each square. At any step, Kelvin the Frog chooses two adjacent squares (two squares are adjacent if they share a side) and increments the numbers in both of them by 1. Can Kelvin make every square contain a different power of 2? (c) A tournament consists of single games between every pair of players, where each game has a winner and loser with no ties. A set of people is dominated if there exists a player who beats all of them. Does there exist a tournament in which every set of 2017 people is dominated? (d) Every cell of a 19×1919 \times 19 grid is colored either red, yellow, green, or blue. Does there necessarily exist a rectangle whose sides are parallel to the grid, all of whose vertices are the same color? (e) Does there exist a c∈R+c \in \mathbb{R}^{+}such that max(|A⋅A|,|A+A|)≥c|A|log2|A|\max (|A \cdot A|,|A+A|) \geq c|A| \log ^{2}|A| for all finite sets A⊂ZA \subset \mathbb{Z}? (f) Can the set {1,2,…,1093}\{1,2, \ldots, 1093\} be partitioned into 7 subsets such that each subset is sum-free (i.e. no subset contains a,b,ca, b, c with a+b=c)a+b=c)?
Plain-text mathematical notation (without MathML)
(a) Can 1000 queens be placed on a 2017×2017 chessboard such that every square is attacked by some queen? A square is attacked by a queen if it lies in the same row, column, or diagonal as the queen. (b) A 2017×2017 grid of squares originally contains a 0 in each square. At any step, Kelvin the Frog chooses two adjacent squares (two squares are adjacent if they share a side) and increments the numbers in both of them by 1. Can Kelvin make every square contain a different power of 2? (c) A tournament consists of single games between every pair of players, where each game has a winner and loser with no ties. A set of people is dominated if there exists a player who beats all of them. Does there exist a tournament in which every set of 2017 people is dominated? (d) Every cell of a 19×19 grid is colored either red, yellow, green, or blue. Does there necessarily exist a rectangle whose sides are parallel to the grid, all of whose vertices are the same color? (e) Does there exist a c∈R^(+)such that max(|A⋅A|,|A+A|)≥c|A|log²|A| for all finite sets A⊂Z? (f) Can the set {1,2,…,1093} be partitioned into 7 subsets such that each subset is sum-free (i.e. no subset contains a,b,c with a+b=c)?
Original LaTeX notation
(a) Can 1000 queens be placed on a $2017 \times 2017$ chessboard such that every square is attacked by some queen? A square is attacked by a queen if it lies in the same row, column, or diagonal as the queen. (b) A $2017 \times 2017$ grid of squares originally contains a 0 in each square. At any step, Kelvin the Frog chooses two adjacent squares (two squares are adjacent if they share a side) and increments the numbers in both of them by 1. Can Kelvin make every square contain a different power of 2? (c) A tournament consists of single games between every pair of players, where each game has a winner and loser with no ties. A set of people is dominated if there exists a player who beats all of them. Does there exist a tournament in which every set of 2017 people is dominated? (d) Every cell of a $19 \times 19$ grid is colored either red, yellow, green, or blue. Does there necessarily exist a rectangle whose sides are parallel to the grid, all of whose vertices are the same color? (e) Does there exist a $c \in \mathbb{R}^{+}$such that $\max (|A \cdot A|,|A+A|) \geq c|A| \log ^{2}|A|$ for all finite sets $A \subset \mathbb{Z}$? (f) Can the set $\{1,2, \ldots, 1093\}$ be partitioned into 7 subsets such that each subset is sum-free (i.e. no subset contains $a, b, c$ with $a+b=c)$?

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task_id
571ff1f5-9b48-5a44-960e-83e1e19062cf
task_key
test--571ff1f5-9b48-5a44-960e-83e1e19062cf
task_revision_id
1
config / split
/ test

Official source

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