Problem 4

Part of 2005 Federal Math Competition of S&M

Problems(3)

parameters guaranteeing a winning strategy (Serbia MO 2005 1st Grade P4)

Source:

c

p

b

white balls on a table. Two players

A

B

play a game by alternately making moves. In every move, a player takes two or three balls from the table. Player

A

begins. A player wins if after his/her move at least one of the three colors no longer exists among the balls remaining on the table. For which values of

c,p,b

A

have a winning strategy?

gamecombinatorics

area of spots inside a circle

Source: Serbia MO 2005 2nd Grade P4

Inside a circle

k

R

some round spots are made. The area of each spot is

1

. Every radius of circle

k

, as well as every circle concentric with

k

, meets in no more than one spot. Prove that the total area of all the spots is less than

\pi\sqrt R+\frac12R\sqrt R.

circlegeometryarea

markers on chessboard, removing markers by a rule

Source: Serbia MO 2005 3&4th Grades P4

On each cell of a

2005\times2005

chessboard, there is a marker. In each move, we are allowed to remove a marker that is a neighbor to an even number of markers (but at least one). Two markers are considered neighboring if their cells share a vertex.
(a) Find the least number

n

of markers that we can end up with on the chessboard. (b) If we end up with this minimum number

n

of markers, prove that no two of them will be neighboring.

gamecombinatorics