KöMaL 2012 B4430

We play the following game of tossing a coin: We make a bet before each toss. The bet may be at most as large as all the money we have. Then we make a guess on the result. If our guess is right, we will receive the double of the bet, otherwise we will lose it. Determine those values of $k$>1 for which there exists an integer $n$with a strategy such that the probability of receiving $k$ times our initial amount of money in at most $n$ games is at least $\frac{1}{k}$ .