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There exist a increasing sequence

Source: 2012 China Mathematical Olympaid P3

January 13, 2012

ceiling functionalgebra proposedalgebra

Problem Statement

Prove for any

M>2

, there exists an increasing sequence of positive integers

a_1<a_2<\ldots

satisfying: 1)

a_i>M^i

for any

i

; 2) There exists a positive integer

m

and

b_1,b_2,\ldots ,b_m\in\left\{ -1,1\right\}

, satisfying

n=a_1b_1+a_2b_2+\ldots +a_mb_m

if and only if

n\in\mathbb{Z}/ \{0\}