KöMaL 2017 B4901

An epidemic broke out in Smurf village after a few residents contracted a disease. Luckily, every smurf recovers from the illness in one day, and then they will be immune to the disease for another day. However, from the following day onwards they may catch the disease again. The transition between the sick and healthy states always occurs at night when the smurfs are sleeping. Unfortunately, the smurfs never give up their habit of visiting all their friends every day, not even when they are ill. So when a sick smurf meets a healthy but not immune one, the latter will inevitably catch the disease. Given that Smurf village has 100 inhabitants, show that the epidemic will necessarily terminate by the 101th day following the outbreak.