OBM 2017 Problem 1

For each real number $r$ between $0$ and $1$ we can represent $r$ as an infinite decimal $r = 0.r_1r_2r_3\dots$ with $0 \leq r_i \leq 9$ . For example, $\frac{1}{4} = 0.25000\dots$ , $\frac{1}{3} = 0.333\dots$ and $\frac{1}{\sqrt{2}} = 0.707106\dots$ .

a) Show that we can choose two rational numbers $p$ and $q$ between $0$ and $1$ such that, from their decimal representations $p = 0.p_1p_2p_3\dots$ and $q = 0.q_1q_2q_3\dots$ , it's possible to construct an irrational number $\alpha = 0.a_1a_2a_3\dots$ such that, for each $i = 1, 2, 3, \dots$ , we have $a_i = p_1$ or $a_1 = q_i$ .

b) Show that there's a rational number $s = 0.s_1s_2s_3\dots$ and an irrational number $\beta = 0.b_1b_2b_3\dots$ such that, for all $N \geq 2017$ , the number of indexes $1 \leq i \leq N$ satisfying $s_i \neq b_i$ is less than or equal to $\frac{N}{2017}$ .