MathDB

5

Part of 2005 May Olympiad

Problems(2)

battleship revisited

Source: XI May Olympiad (Olimpiada de Mayo) 2005 L2 P5

9/19/2022
The enemy ship has landed on a 9×99\times 9 board that covers exactly 55 squares of the board, like this: https://cdn.artofproblemsolving.com/attachments/2/4/ae5aa95f5bb5e113fd5e25931a2bf8eb872dbe.png The ship is invisible. Each defensive missile covers exactly one square, and destroys the ship if it hits one of the 55 squares that it occupies. Determine the minimum number of missiles needed to destroy the enemy ship with certainty .
combinatoricscombinatorial geometry
1-7 at each box of 7x7 board

Source: XI May Olympiad (Olimpiada de Mayo) 2005 L1 P5

9/22/2022
a) In each box of a 7×77\times 7 board one of the numbers is written: 1,2,3,4,5,61, 2, 3, 4, 5, 6 or 77 of so that each number is written in seven different boxes. Is it possible that in no row and no column are consecutive numbers written?
b) In each box of a 5×55\times 5 board one of the numbers is written: 1,2,3,41, 2, 3, 4 or 55 of so that each one is written in five different boxes. Is it possible that in no row and in no column are consecutive numbers written?
combinatorics