Estonian Math Competitions

Let $x_i$ be the number written on the $i$th line, then for all $i = 1, 2, \dots$ we have $x_{i+1} = x_i^2 - 66x_i + 1122$. Notice that this is equivalent to $x_{i+1} - 33 = x_i^2 - 66x_i + 1089 = (x_i - 33)^2$. Denoting $a_n = x_n - 33$, we acquire $a_{i+1} = a_i^2$ for all $i = 1, 2, \dots$, which means that $a_1 = 43 - 33 = 10$, $a_2 = a_1^2 = 10^2$, $a_3 = a_2^2 = 10^4$ and in general $a_i = 10^{2^{i-1}}$. So $x_{2021} = a_{2021} + 33 = 10^{2^{2020}} + 33$.