M15-02

Official Solution:


If \(m\) and \(n\) are positive numbers, is \(\frac{mn}{m + n} > 1\)?

Let's simplify the question. Since both \(m\) and \(n\) are positive, the question "is \(\frac{mn}{m + n} > 1\)?" can be rephrased as "is \(\frac{m+n}{mn} < 1\)?", and further as "is \(\frac{1}{n} +\frac{1}{m} < 1\)?"

(1) \(\frac{1}{m} \gt \frac{1}{n} > \frac{1}{2}\)

Since both \(\frac{1}{m}\) and \(\frac{1}{n}\) are greater than \(\frac{1}{2}\), their sum is greater than 1: \(\frac{1}{n} +\frac{1}{m} > 1\). Therefore, we have a NO answer to the question. Sufficient.

(2) \(m = n - 1\)

If both \(m\) and \(n\) are large enough numbers, such as 99 and 100, then \(\frac{1}{n} +\frac{1}{m} < 1\), giving a YES answer to the question. However, if \(m = 1\) and \(n = 2\), then \(\frac{1}{n} +\frac{1}{m} > 1\), giving a NO answer to the question. Not sufficient.


Answer: A