KZ-MO 2004 Problem 4

In some village there are $1000$ inhabitants. Every day, each of them shares the news they learned yesterday with all their friends. It is known that any news becomes known to all residents of the village. Prove that it is possible to select $90$ residents so that if you tell all of them at the same time some news, then in $10$ days it will become known to all residents of the village.