MathDB

10

Part of 2019 CHMMC (Fall)

Problems(1)

2019 Fall Team #10

Source:

4/17/2022
nn players are playing a game. Each player has nn tokens. Every turn, two players with at least one token are randomly selected. The player with less tokens gives one token to the player with more tokens. If both players have the same number of tokens, a coin flip decides which player receives a token and which player gives a token. The game ends when one player has all the tokens. If n=2019n = 2019, suppose the maximum number of turns the game could take to end can be written as 1d(a⋅20193+b⋅20192+c⋅2019)\frac{1}{d} (a \cdot 2019^3 + b \cdot 2019^2 + c \cdot 2019) for integers a,b,c,da, b, c, d. Find abcd\frac{abc}{d} .
combinatorics