1. On the board in the laboratory, two numbers are written. Every day, senior researcher Petya erases both numbers from the board and writes down their arithmetic mean and harmonic mean instead. On 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 1999th day. (The arithmetic mean of two numbers $a$ and $b$ is the number $\frac{a+b}{2}$, and the harmonic mean is the number $\frac{2}{1 / a+1 / b}$.)
1. The product of the numbers on the board does not change. Indeed,

$$
\frac{a+b}{2} \cdot \frac{2}{1 / a+1 / b}=\frac{a+b}{2} \cdot \frac{2 a b}{a+b}=a b
$$

Therefore, on the 1999th day, the product will be the same as it was on the first day. See also fact 2.