Problem 5

Let

a_1,a_2,\ldots

be an infinite sequence of positive integers. Suppose there is an integer

N>1

such that, for every

n\ge N

, the number

\frac{a_1}{a_2}+\frac{a_2}{a_3}+\cdots+\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}

is an integer. Prove that there is a positive integer

M

such that

a_m=a_{m+1}

for all

m\ge M

Step 1 of 8: Take consecutive differences

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In plain words

The long cyclic sums differ in only three terms.

S_n=\sum_{i=1}^{n-1}\frac{a_i}{a_{i+1}}+\frac{a_n}{a_1}

Detailed analysis

For $n>N$ , both $S_n$ and $S_{n+1}$ are integers. Their difference is $T_n=S_{n+1}-S_n=\frac{a_{n+1}}{a_1}-\frac{a_n}{a_1}+\frac{a_n}{a_{n+1}}$ , so $T_n\in\mathbb Z$ .