9.2. On the board in the laboratory, two numbers are written. Every day, the senior researcher Pyotr Ivanovich erases both numbers from the board and writes down their arithmetic mean and harmonic mean ${ }^{2}$. In the morning of the first day, the numbers 1 and 2 were written on the board.

Find the product of the numbers written on the board in the evening of the 2016th day.
# Solution:

The product of the numbers on the board does not change.

Indeed, $\frac{a+b}{2} \times \frac{2}{\frac{1}{a}+\frac{1}{b}}=a b$. Therefore, the desired product is 2.