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fractions, integers

Source: Tuymaada 2005, Day 2, Problem 6

July 30, 2005

algebra unsolvedalgebra

Problem Statement

Given are a positive integer

n

and an infinite sequence of proper fractions

x_0 = \frac{a_0}{n}

,

\ldots

,

x_i=\frac{a_i}{n+i}

, with

a_i < n+i

. Prove that there exist a positive integer

k

and integers

c_1

,

\ldots

,

c_k

such that

c_1 x_1 + \ldots + c_k x_k = 1.

Proposed by M. Dubashinsky

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