Alenka received a piece of paper with the following message:

A. At most one of the statements $A, B, C, D, E$ is true.

B.

C. All statements $A, B, C, D, E$ are true.
D.

E. Statement A is true.

Statements $B$ and $D$ were written in invisible ink, which can only be read under a special lamp. Before Alenka found such a lamp, she managed to decide whether she could trust these statements.

Determine which of the statements $A, B, C, D, E$ are true and which are false.

(I. Jančigová)

Hint. Proceed systematically and consider all consequences.
If statement $A$ were true, then statement $E$ would also be true. However, this would mean two statements are true, which would contradict statement $A$.

If statement $C$ were true, then all statements would be true. However, this would mean more than one true statement, which would contradict statement $A$.

If statement $E$ were true, then statement $A$ would also be true. However, this would mean two true statements, which would contradict statement $A$.

Therefore, all visible statements are false. From the falsity of $A$, it follows that at least two of all statements are true, and these must be the invisible ones.

Statements $B, D$ are true, statements $A, C, E$ are false.

Note. In the above, we tacitly assume that each statement is either true or false (technically, we speak of propositions). Be aware that there are problematic statements whose truth or falsity cannot be determined (see, for example, This statement is false).