M15-37

Official Solution:


If \(a^2 - b^2 = b^2 - c^2\), is \(a = |b|\)?

(1) \(b = |c|\).

Squaring the above gives \(b^2 = c^2\). Substituting this into the equation given in the stem yields:

\(a^2 - b^2 = b^2 - b^2\);

\(a^2 - b^2 = 0\);

\(a^2 = b^2\);

\(|a| = |b|\).

If \(a\) is non-negative, then \(|a| = |b|\) implies \(a = |b|\). However, if \(a\) is negative, then \(|a| = |b|\) implies \(-a = |b|\). For example, if \(a = b = c = 1\), the answer is YES; but if \(a = -1\) and \(b = c = 1\), the answer is NO. Not sufficient.

(2) \(b = |a|\).

The above implies that \(b\) is non-negative, hence \(|b| = b\), and the question becomes whether \(a=b\). If \(a\) is also non-negative, then \(b = |a|\) gives \(a=b\), and in this case, the answer to the question is YES. However, if \(a\) is negative, then \(b=|a|\) gives \(b=-a\), and in this case, the answer to the question is NO. For example, if \(a=b=c=1\), the answer to the question is YES, but if \(a=-1\) and \(b=c=1\), the answer to the question is NO. Not sufficient.

(1)+(2) When combining the statements, we know that \(|a| = |b| = |c|\), with \(b\) being non-negative. We can rephrase the question as: is \(a=b\)? The issue is that we still don't know the sign of \(a\). If \(a\) is also non-negative, the answer to the question is YES. However, if \(a\) is negative, the answer to the question is NO. We can consider the same examples as above: if \(a=b=c=1\), the answer to the question is YES, but if \(a=-1\) and \(b=c=1\), the answer to the question is NO. Not sufficient.


Answer: E