# 1986 AJHSME Problems/Problem 22

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## Problem

Alan, Beth, Carlos, and Diana were discussing their possible grades in mathematics class this grading period. Alan said, "If I get an A, then Beth will get an A." Beth said, "If I get an A, then Carlos will get an A." Carlos said, "If I get an A, then Diana will get an A." All of these statements were true, but only two of the students received an A. Which two received A's?

$\text{(A)}\ \text{Alan, Beth} \qquad \text{(B)}\ \text{Beth, Carlos} \qquad \text{(C)}\ \text{Carlos, Diana}$

$\text{(D)}\ \text{Alan, Diana} \qquad \text{(E)}\ \text{Beth, Diana}$

## Solution

Let's say that Alan gets an A. Well, from his statement, then Beth would also get an A. But from her statement, Carlos would get an A. And from his statement, Diana would also get an A. So all 4 would get A's, but the problem said only 2 got A's.

Let's say that Beth gets an A. From her statement, we know that Carlos get an A, and from his statement we know that Diana gets an A. But that makes 3, which is not 2.

If Carlos gets an A, then Diana gets an A. That makes 2, so $\boxed{\text{C}}$ is the right answer. Note that although Beth said "If I get an A, then Carlos will get an A.", that does NOT mean that "If Carlos gets an A, then I will get an A."