The main observation in this problem is this: as soon as a player guesses correctly how the coin will land, they can guess incorrectly in all remaining games. Therefore, all bets after a win should be $1.

To begin, let's show that the player will not be able to leave having lost $1 or less. Let's see what they would have to do for that.

1. The first time, the player will have to bet no less than $3, because they could win this game and lose the next 4.
2. If they lost the first game, then in the second, they would have to bet no less than $5: because they might win only this round, but lose at least $3 on the previous one and at least $3 on the next three.
3. If they lost the second game as well, then on the third, their bet should be at least $9: if they win, they need to compensate for at least $10 in losses (3 for the first game, 5 for the second, 1 for the fourth, and 1 for the fifth).
4. If they lost the third game as well, then on the fourth, they would have to bet at least $17: a win should compensate for at least $3 + 5 + 9 + 1 = 18 in losses.
5. If they lost the fourth game as well, then they have already lost at least $3 + 5 + 9 + 17 = 34, and even the maximum bet of $17 cannot make up for the loss.

Now let's prove that the player can bet in such a way as to leave having lost only $2. Similarly to the above reasoning, we understand that as long as the player is losing, they should bet $2, then $3, $5, $9, and $17, and as soon as they win, all subsequent rounds should be bets of $1. It is easy to check that in every situation, the player will lose no more than $2.

Comment. Note that if the player is allowed to bet no more than 5 times, they can "end up in the plus." For this, they need to bet $1, $2, $4, $8, and $16 and leave after they win: then they will definitely win $1. The proof that in this case, they cannot win more is similar to the proof in the solution.

## Answer

98.

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Authors: Binnov A.D., Binkov Yu.A., Gorskaya E.S.

A casino offers a game with the following rules. A player bets any whole number of dollars (but no more than they currently have) on either heads or tails. Then a coin is flipped. If the player guesses correctly how it will land, they get back their bet and an equal amount of money as a reward. If they guess incorrectly, the casino takes their bet. If a player is unlucky four times in a row, the casino awards them a consolation win in the next game, regardless of how the coin lands. Joe came to the casino with 100 dollars. He committed to making exactly five bets and never betting more than 17 dollars. What is the largest amount of money he can guarantee to take away from the casino after such a game?