XXIX Rioplatense Mathematical Olympiad

Notice that $9L^2+2L=(9L+2)L$. The number $9L$ is the 2022-digit number made only by nines. By adding 2 we get $9L+2=\underset{2022\text{ digits}}{\underbrace{1000\dots 01}}=10^{2022}+1$. Now it is clear that $$\begin{gathered} (9L+2)L=10^{2022}L+L=\underset{2022\text{ digits}}{\underbrace{111\dots 11}}\underset{2022\text{ digits}}{\underbrace{000\dots 00}}+\underset{2022\text{ digits}}{\underbrace{111\dots 11}} \\ =\underset{4044\text{ digits}}{\underbrace{1111\dots 111}}. \end{gathered}$$ Hence the answer is 4044.