Estonian Math Competitions

Question

On the first line of a notebook Juku writes the number 43. On every following line he writes the number x^2 - 66x + 1122, where x is the number on the previous line. Find the number that Juku will write on the 2021st line.

Accepted Answer

Let x_i be the number written on the ith line, then for all i = 1, 2, 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, , 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.