Question 121
There is a village with a population of $2007$. This village has no name. You are God of this village and you want the villagers to decide the name of this village. Every villager has one idea of the village's name.
Each villager can send a letter to each villager (including himself). And every villager can send any number of letters every day. Letters are collected in the evening and delivered all at once the next morning every day. The villager who sends a letter can decide to whom the letter should be delivered. And each villager can send a letter to tell God his idea of the name of the village only one time. This idea does not need to be the same as the idea which he and the other villagers had thought of at first. And every villager's action is only writing a letter.
Every villager can be classified as an honest person or a liar. You and every villager do not know who is an honest person and who is a liar. But you know that the number of liars is less than or equal to $T$, and there is at least one honest person in this village.
You can give instructions to every villager only once, at noon of one day. An honest person necessarily follows the instruction, but you do not know whether a liar follows the instruction. Find the maximum $T$ for which there exists an instruction which fulfills the conditions below.
* At last, every honest person sends a letter to God, and every honest person sends the same idea of the village's name.
* If every honest person had thought of the same idea of the name of the village at first, every honest person sends this idea to God.
Each villager can send a letter to each villager (including himself). And every villager can send any number of letters every day. Letters are collected in the evening and delivered all at once the next morning every day. The villager who sends a letter can decide to whom the letter should be delivered. And each villager can send a letter to tell God his idea of the name of the village only one time. This idea does not need to be the same as the idea which he and the other villagers had thought of at first. And every villager's action is only writing a letter.
Every villager can be classified as an honest person or a liar. You and every villager do not know who is an honest person and who is a liar. But you know that the number of liars is less than or equal to $T$, and there is at least one honest person in this village.
You can give instructions to every villager only once, at noon of one day. An honest person necessarily follows the instruction, but you do not know whether a liar follows the instruction. Find the maximum $T$ for which there exists an instruction which fulfills the conditions below.
* At last, every honest person sends a letter to God, and every honest person sends the same idea of the village's name.
* If every honest person had thought of the same idea of the name of the village at first, every honest person sends this idea to God.