1. On the board, 2020 quadratic equations are written:

$$
\begin{gathered}
2020 x^{2} + b x + 2021 = 0 \\
2019 x^{2} + b x + 2020 = 0 \\
2018 x^{2} + b x + 2019 = 0 \\
\ldots \\
x^{2} + b x + 2 = 0
\end{gathered}
$$

(each subsequent equation is obtained from the previous one by decreasing the leading coefficient and the constant term by one). 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 $\frac{2020}{2019}$, the third is $-\frac{2019}{2018}, \ldots$, the two thousand and twentieth is $\frac{2}{1}$. Therefore, the product of the roots of all equations is $\frac{2021}{2020} \cdot \frac{2020}{2019} \cdot \frac{2019}{2018} \cdot \frac{2018}{2017} \cdot \ldots \cdot \frac{3}{2} \cdot \frac{2}{1}=2021$. Answer: 2021.

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