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Miklós Schweitzer 1985, Problem 8

Source: Miklós Schweitzer 1985

September 5, 2016

Miklos Schweitzercollege contestsSequencesdiscrepancy theory

Problem Statement

Let

\frac{2}{\sqrt5+1}\leq p < 1

, and let the real sequence

\{ a_n \}

have the following property: for every sequence

\{ e_n \}

of

0

's and

\pm 1

's for which

\sum_{n=1}^\infty e_np^n=0

, we also have

\sum_{n=1}^\infty e_na_n=0

. Prove that there is a number

c

such that

a_n=cp^n

for all

n

. [Z. Daroczy, I. Katai]

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