MathDB

Welcome to the USAYNO, where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer

n

problems and get them all correct, you will receive

\max(0, (n-1)(n-2))

points. If any of them are wrong (or you leave them all blank), you will receive

0

points.
Your answer should be a six-character string containing 'Y' (for yes), 'N' (for no), or 'B' (for blank). For instance if you think 1,2, and 6 are 'yes' and 3 and 4 are 'no', you should answer YYNNBY (and receive

12

points if all five answers are correct, 0 points if any are wrong).
(a) Can

1000

queens be placed on a

2017\times2017

chessboard such that every square is attacked by some queen? A square is attacked by a queen if it lies on the same row, column, or diagonal as the queen.
(b) A

2017\times2017

grid of squares originally contains a

0

in each square. At any step, Kelvin the Frog choose 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 a 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\times19

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|)\ge c|A|\log^2|A|

for all finite sets

A\subset \mathbb{Z}

?
(f) Can the set

\{1, 2, \dots, 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

)?
[color = red]The USAYNO disclaimer is only included in problem 33. I have included it here for convenience.

Problem Statement