A question of Logic in Olympiad A question has appeared in our Informatics Olympiad which there is a lot of discussion over it. the problem states:
Below are 5 statements. At most how many of them can be true together?
a) if b is true then this statement is false.
b) if number of the true statements is greater than $2$, then one of them is c.
c) at least one of a and d is false.
d) b and c are both true or both false.
e) b is true or false.  
many say this number is $3$, by statements a,d,e being true and the rest false, and 
many other say this number is $4$, a being false and the rest being true. what is your opinion? which group say the truth?
EDIT. I made a mistake and typed ''wrong'' in statement c instead of ''false''. now it's correct.
 A: a cannot be false: for a to be false you need b to be true and a to not be false; that is, b and a must both be true. That means that
$$\mathbf{a}\text{ false}\Rightarrow \mathbf{a}\text{ true}.$$
That means that a will be true, since $(\neg P\to P)\to P$ is a tautology.
In particular, it is impossible for a to be false and all other statements to be true. 
Also, e is always true. So at least a and e are always true. 
Corrected (misread b in my first pass).
Since a is true, by contrapositive it follows that b is false. So we must have that there are more than two true statements, but c is false. Since b and c are both false, then d is true (which is our third true statement).
So the answer is 3 statements.

No longer applicable:
If, as magma suggests, "wrong" should not be taken as a synonym for "false", but rather as saying something like "ill-formed", "self-referential", "cannot be assigned a truth-value", etc., then you could make a case for saying that a is self-referential, hence "wrong", while the other four statements are true. But if that is the case, then the answer you quote, "a false and the rest true" would also be incorrect, since the conclusion is not that "a is false", but rather that "a is wrong". 
A: statement a is self-referential. Normally in propositional logic you do not admit self-reference because it leads to antinomies. The statement "this statement is false" is neither true nor false.  It is wrong.
So c is true (since a is wrong).
Since c is true, the consequent in b ("one of them is c") is true, so b is true.
So d is also true and....
e is true. 
Answer: b,c,d and e are true, a is wrong
Please note: a is not false, it is wrong. This is what the judges wanted the competitors to realize, as you see in the wording of c
