[Mathematical logic (miscellaneous).]

Once, a strange notebook was found on a staircase. It contained a hundred statements:

"In this notebook, there is exactly one false statement";

"In this notebook, there are exactly two false statements";

"In this notebook, there are exactly three false statements";

"In this notebook, there are exactly a hundred false statements."

Are there any true statements among these, and if so, which ones?
Notice that in the notebook, there are one hundred statements, each two of which contradict each other.

## Solution

The fact that there are 100 statements in the notebook, each two of which contradict each other, means that if there are any true statements among them, there cannot be more than one. Let's see if there can be at least one true statement. If exactly one statement is true, then exactly ninety-nine are false. And there is such a statement in the notebook: "In this notebook, there are exactly ninety-nine false statements." Therefore, there is exactly one true statement in the notebook.

## Answer

One true statement: "In this notebook, there are exactly ninety-nine false statements."