Ana and Benito play a game consisting of 2020 rounds. Initially, there are 2020 cards on the table, numbered from 1 to 2020, and Ana has an additional card with the number 0. In round , the player who does not have card decides whether to take card or hand it over to the other player. The number on each card indicates its value in points. When the game ends, whoever has more points wins. Determine which player has a winning strategy, or whether both players can force a tie, and describe the strategy to be followed.
Problem 1399
Official solution
Both players can force a tie. We divide the game into 505 stages, each with four consecutive rounds of the form
We will show that each player can secure at least half of the points of each stage, regardless of which player has card . It does not matter what happens in round . From there:
* The player who receives card (without loss of generality, we may assume it is Ana) can secure at least a tie in the 4-turn stage. Indeed, if Benito also hands her card and card , then Ana has already received points and wins the 4-turn stage. If Benito hands Ana card and keeps card , Ana takes card and wins the 4-turn stage. If Benito keeps card , Ana hands card to Benito and keeps card , thereby tying the 4-turn stage. In summary, whoever receives card can always secure at least a tie.
* The player who does not receive card (without loss of generality, Benito) keeps card . If Ana hands him card , Benito already has points and guarantees a tie in the 4-turn stage. If Ana keeps card , Benito keeps card , thereby ending with points and winning the 4-turn stage. In summary, whoever does not receive card can always secure at least a tie.