N.T.TUAN wrote: Let $ n > 2$ be an integer. Suppose that $ a_{1},a_{2},...,a_{n}$ are real numbers such that $ k_{i} = \frac {a_{i - 1} + a_{i + 1}}{a_{i}}$ is a positive integer for all $ i$(Here $ a_{0} = a_{n},a_{n + 1} = a_{1}$). Prove that $ 2n\leq a_{1} + a_{2} + ... + a_{n}\leq 3n$. $ k_1+...+k_n =\sum^{n}_{i=1} \frac{a_i}{a_{i+1}}+\frac{a_{i+1}}{a_i}$ On the other side,we will actually show that $ k_1+...+k_n \leq 3n-2$ for $ n \geq 2$,by induction on $ n$

@N.T.TUAN, i solved it before and also saw it in book "Andreescu - Contests Around the World 1997-1998.pdf"

Taiwan 2014 TST 3 Quiz 3, P1