4
Part of 2007 May Olympiad
Problems(2)
Prime numbers
Source: May Olympiad(Olimpiada de Mayo) 2007
2/2/2018
Alex and Bruno play the following game: each one, in your turn, the player writes, exactly one digit, in the right of the last number written. The game finishes if we have a number with digits( distincts ) and Alex starts the game. Bruno wins if the number with digits is a prime number, otherwise Alex wins.
Which player has the winning strategy?
combinatoricsnumber theoryprime numbers
one lamp at each square of a 7x7 board
Source: XIII May Olympiad (Olimpiada de Mayo) 2007 L1 P4
9/22/2022
A board has a lamp on each of its squares, which can be on or off.
The allowed operation is to choose consecutive cells of a row or a column that have two lamps neighboring each other on and the other off, and change the state of all three. Namely
https://cdn.artofproblemsolving.com/attachments/e/b/28737b19c940ff5e1c98d05533c77069e990f5.png
Give a configuration of exactly lit lamps located in the first rows of the board such that, through a succession of permitted operations, a single lamp is lit on the board and that it is located in the last row. Show the sequence of operations used to achieve the goal.
combinatorics