MathDB

For positive integers

t,a,b,

(t,a,b)

-game is a two player game defined by the following rules. Initially, the number

t

is written on a blackboard. At his first move, the 1st player replaces

t

with either

t-a

t-b

. Then, the 2nd player subtracts either

a

b

from this number, and writes the result on the blackboard, erasing the old number. After this, the first player once again erases either

a

b

from the number written on the blackboard, and so on. The player who first reaches a negative number loses the game. Prove that there exist infinitely many values of

t

for which the first player has a winning strategy for all pairs

(a,b)

with

a+b=2005

Problem Statement