Everyone Is Pointing At Each Other

Hack:

Because none of the options say that Statement 5 is true, it must be false. This means Statement 3 is true. Among the options, there is only one option that says two correct statements.

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Actual solution:

Note that Statement X saying that "Statement Y is false" is equivalent to the known true statement "either Statement X or Statement Y is true, but not both". Thus we know that either Statement 3 or Statement 5 is true, but not both.

Less well-known is that Statement X saying that "Statement Y and Statement Z are both true or both false" is equivalent to the known true statement "an odd number of Statements X, Y, Z are true".

From Statement 2, we can conclude that an odd number of Statements 2, 3, 5 are true. Since Statements 3 and 5 contribute exactly one true statement, Statement 2 must be false.

From Statement 4, we can conclude that "an odd number of Statements 1, 2, 4 are true". Since Statement 2 is false, this becomes "either Statement 1 or Statement 4 is true, but not both".

Thus Statement 2 is false; one of Statements 1, 4 is true; one of Statements 3, 5 is true. This means there are two true statements, so Statement 3 is true and Statement 5 is false.

Now, Statement 1 says that "Statements 2 and 5 have the same truth value". Since Statement 2 and Statement 5 are both false, this means Statement 1 is true. Since only one of Statements 1, 4 is true, we have Statement 4 to be false.

This means Statements 1 and 3 are the only true statements.

For the second option, 2,3,4 cannot be true at the same time, because statement 3 says only 2 are true, if including itself, there will be 3 true statements, so second option is eliminated. It also applies to the last option, statement 3 cannot come alone since it says there are 2 true statements, so last option is eliminated. For the third option, it says only 2, let's look at statement 2 , statements 3 and 5 cannot be true at the same time, because if statement 5 is true, statement 3 must be wrong. Same for the other situation, if 5 is false, 3 cannot be false because otherwise, 5 will be true, so they cannot be both false. After eliminating the third option, it leaves the first option as the answer

Read statement 3 and statement 5 it gave away the answer