MathDB

On a party with

99

guests, hosts Ann and Bob play a game (the hosts are not regarded as guests). There are

99

chairs arranged in a circle; initially, all guests hang around those chairs. The hosts take turns alternately. By a turn, a host orders any standing guest to sit on an unoccupied chair

c

. If some chair adjacent to

c

is already occupied, the same host orders one guest on such chair to stand up (if both chairs adjacent to

c

are occupied, the host chooses exactly one of them). All orders are carried out immediately. Ann makes the first move; her goal is to fulfill, after some move of hers, that at least

k

chairs are occupied. Determine the largest

k

for which Ann can reach the goal, regardless of Bob's play.

5

Problems(1)