1. On the board, three quadratic equations are written:

$$
\begin{gathered}
2020 x^{2}+b x+2021=0 \\
2019 x^{2}+b x+2020=0 \\
x^{2}+b x+2019=0
\end{gathered}
$$

Find the product of the roots of all the equations written on the board, given that each of them has two real roots.
Solution. According to the theorem converse to Vieta's theorem, the product of the roots of the first equation is $-\frac{2021}{2020}$, the product of the roots of the second equation is $\frac{2020}{2019}$, and the third is 2019. Therefore, the product of the roots of all equations is $\frac{2021}{2020} \cdot \frac{2020}{2019} \cdot 2019=2021$. Answer: 2021.

Comment. The statement of the theorem converse to Vieta's theorem is mistakenly called Vieta's theorem by the olympiad participant - no points should be deducted.