MathDB

Two sequences

Source: Iran MO 3rd round 2019 mid-terms - Number theory P1

August 1, 2019

number theory

Problem Statement

Given a number

k\in \mathbb{N}

.

\{a_{n}\}_{n\geq 0}

and

\{b_{n}\}_{n\geq 0}

are two sequences of positive integers that

a_{i},b_{i}\in \{1,2,\cdots,9\}

. For all

n\geq 0

\left.\overline{a_{n}\cdots a_{1}a_{0}}+k \ \middle| \ \overline{b_{n}\cdots b_{1}b_{0}}+k \right. .

Prove that there is a number

1\leq t \leq 9

and

N\in \mathbb{N}

such that

b_n=ta_n

for all

n\geq N

.\\ (Note that

(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0

)

Back to Problems View on AoPS