Bariz - Problem

Source: 2018 Centroamerican and Caribbean Math Olympiad

Tags: Centroamerican, combinatorics, game strategy

jestrada

25.06.2018 20:09

There are 2018 cards numbered from 1 to 2018. The numbers of the cards are visible at all times. Tito and Pepe play a game. Starting with Tito, they take turns picking cards until they're finished. Then each player sums the numbers on his cards and whoever has an even sum wins. Determine which player has a winning strategy and describe it. P.S. Proposed by yours truly

Tito has a winning strategy. In order to win, Tito has to get an even amount of cards that are even. Firstly, Tito chooses an odd. Then, whatever Pepe chooses, Tito follows him. As there are 1009 evens and 1008 odds, if Tito cannot follow, Pepe chooses 505 evens and k odds. But, Tito chooses 504 evens and there has no more even left. So, Tito has chosen an even amount of cards that are even. Line 5 and line 6 provide us that Tito can follow Pepe's choose. We just focus on the odd part. If Tito can follow Pepe's choose, including the first choose of Tito, Tito has chosen an odd amount of cards that are odd. On the other words, Tito gets an even amount of cards that are even which means that we are done with the proof.