1999 AMC 12 Problem 10

Question

A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false.

I. The digit is

1.

II. The digit is not

2.

III. The digit is

3.

IV. The digit is not

4.

Which one of the following must necessarily be correct?

\mathrm{I}

is true.

\mathrm{I}

is false.

\mathrm{II}

is true.

\mathrm{III}

is true.

\mathrm{IV}

is false.

Answer: C

Concepts:logical deduction

Difficulty rating: 1370

Small hint:

Statements I and III cannot both be true

Big hint:

Exactly one statement is false, so the false one is I or III

Solution:

Statements I and III cannot both be true, so the single false statement is one of them. Therefore statements II and IV are both true, which makes “II is true” necessarily correct.

The digit is thus

1

or

3.

If it were

1,

then (B) and (D) are false; if it were

3,

then (A) is false; and (E) is always incorrect. Only (C) is guaranteed.

Thus, the correct answer is C.

1999 AMC 12 Problem 10

Answer: C

Difficulty rating: 1370

Small hint:

Big hint:

Solution:

Problem 10 in other years