There are hundred statements.

Statement 1: At least one of these statements is false.
Statement 2: At least two of these statements are false.
Statement 3: At least three of these statements are false.
Statement 4: At least four of these statements are false.

Statement 100: At least a hundred of these statements are false.

What is the maximum number of statements that can be true?

Answer

50 statements are correct and 50 are incorrect. Statements 1 to 50 are true and statements 51 to 100 are false.

If any one of the statements is true, then all of the statements numbered lower than that one must also be true, because the term “at least” is inclusive of a fewer number, and they are numbered in ascending order.

Assuming statement 1 is true; then statement 100 is false.
Assuming statements 1-2 are true; then statements 99-100 are false.
Assuming statements 1-3 are true; then statements 98-100 are false.
Assuming statements 1-4 are true; then statements 97-100 are false.

Assuming statements 1-50 are true; then statements 51-100 are false.

Statement 99 will be the only correct statement if the terms “at least” are replaced with the term “exactly.”, e.g. “Exactly 99 of these statements are false.”

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