A1. The arithmetic mean of two positive numbers is 65, and their geometric mean is 60. What is the absolute value of the difference between these two numbers?
(A) 10
(B) 20
(C) 30
(D) 40
(E) 50
A1. We have $\frac{a+b}{2}=65$ and $\sqrt{a b}=60$. We can rewrite these as the system of equations $a+b=130$ and $a b=3600$. From the first equation, we express $b=130-a$ and substitute it into the second equation, which we then rearrange to $a^{2}-130 a+3600=0$. The solutions are $a_{1}=40$ and $b_{1}=90$ as well as $a_{2}=90$ and $b_{2}=40$. Therefore, it follows that $|a-b|=50$.