MathDB

1

Part of 2019 Benelux

Problems(2)

Pawns and rooks on a chessboard

Source: Benelux MO 2019 P2

4/28/2019
Pawns and rooks are placed on a 2019×20192019\times 2019 chessboard, with at most one piece on each of the 201922019^2 squares. A rook can see another rook if they are in the same row or column and all squares between them are empty. What is the maximal number pp for which pp pawns and p+2019p+2019 rooks can be placed on the chessboard in such a way that no two rooks can see each other?
combinatorics
Four variables ineq

Source: Benelux MO 2019 P1

4/28/2019
[*]Let a,b,c,da,b,c,d be real numbers with 0⩽a,b,c,d⩽10\leqslant a,b,c,d\leqslant 1. Prove that ab(a−b)+bc(b−c)+cd(c−d)+da(d−a)⩽827.ab(a-b)+bc(b-c)+cd(c-d)+da(d-a)\leqslant \frac{8}{27}.[/*] [*]Find all quadruples (a,b,c,d)(a,b,c,d) of real numbers with 0⩽a,b,c,d⩽10\leqslant a,b,c,d\leqslant 1 for which equality holds in the above inequality.
inequalities2019