MathDB

Secco and Ramon are drunk in the real line over the integer points

a

and

b

, respectively. Our real line is a little bit special, though: the interval

(-\infty, 0)

is covered by a sea of lava. Being aware of this fact, and also because they are drunk, they decided to play the following game: initially they choose an integer number

k>1

using a fair dice as large as desired, and therefore they start the game. In the first round, each player writes the point

h

for which it wants to go.
After that, they throw a coin: if the result is heads, they go to the desired points; otherwise, they go to the points

2g - h

, where

g

is the point where each of the players were in the precedent round (that is, in the first round

g = a

for Secco and

g = b

for Ramon). They repeat this procedure in the other rounds, and the game finishes when some of the player is over a point exactly

k

times bigger than the other (if both of the player end up in the point

0

, the game finishes as well).
Determine, in values of

k

, the initial values

a

and

b

such that Secco and Ramon has a winning strategy to finish the game alive.
Observation: If any of the players fall in the lave, he dies and both of them lose the game

Problem Statement