HMMT 2017 U34

(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 $a+b=c)$ ?