[b][u]BdMO National 2015 Secondary Problem 1.[/u][/b]

A crime is committed during the hartal.There are four witnesses.The witnesses are logicians and make the following statement:
Witness [b]One[/b]    said  exactly one  of the four witnesses is a liar.

Witness  [b]Two[/b]  said  exactly  two  of the four witnesses is a liar.

Witness [b]Three[/b]    said  exactly three of the four witnesses is a liar.

Witness  [b]Four[/b]  said  exactly four  of the four witnesses is a liar.

Assume that each of the statements is either true or false.How many of the winesses are liars?

1. Let's denote the witnesses as \( W_1, W_2, W_3, \) and \( W_4 \).
2. We need to determine how many of the witnesses are liars based on their statements:
   - \( W_1 \) says exactly one of the four witnesses is a liar.
   - \( W_2 \) says exactly two of the four witnesses are liars.
   - \( W_3 \) says exactly three of the four witnesses are liars.
   - \( W_4 \) says exactly four of the four witnesses are liars.

3. Let's analyze each statement to see if it can be true:
   - If \( W_1 \) is telling the truth (i.e., exactly one witness is a liar), then the other three witnesses must be telling the truth. However, this creates a contradiction because \( W_2, W_3, \) and \( W_4 \) would all be making false statements, which means more than one witness is a liar. Therefore, \( W_1 \) cannot be telling the truth.
   - If \( W_2 \) is telling the truth (i.e., exactly two witnesses are liars), then two of the witnesses must be telling the truth. However, this creates a contradiction because \( W_1, W_3, \) and \( W_4 \) would all be making false statements, which means more than two witnesses are liars. Therefore, \( W_2 \) cannot be telling the truth.
   - If \( W_3 \) is telling the truth (i.e., exactly three witnesses are liars), then three of the witnesses must be lying. This means only one witness is telling the truth. If \( W_3 \) is the one telling the truth, then \( W_1, W_2, \) and \( W_4 \) must be lying. This scenario is consistent because:
     - \( W_1 \) says exactly one is a liar (false, since three are liars).
     - \( W_2 \) says exactly two are liars (false, since three are liars).
     - \( W_4 \) says exactly four are liars (false, since three are liars).
   - If \( W_4 \) is telling the truth (i.e., exactly four witnesses are liars), then all four witnesses must be lying. This creates a contradiction because if \( W_4 \) is telling the truth, then not all four can be liars. Therefore, \( W_4 \) cannot be telling the truth.

4. From the analysis, the only consistent scenario is when \( W_3 \) is telling the truth and the other three witnesses are lying.

The final answer is \( \boxed{3} \).