### International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria

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### Problem 7

Problem 7. Let $\displaystyle (a_n)_{n=0}^\infty$ be a sequence of real numbers such that $\displaystyle a_0=0$ and

$\displaystyle a_{n+1}^3=a_n^2-8 \quad \text{for} \quad n=0,1,2,\ldots$

Prove that the following series is convergent:

 $\displaystyle \sum_{n=0}^\infty|a_{n+1}-a_n|.$ $\displaystyle (1)$

(Proposed by Orif Ibrogimov, National University of Uzbekistan)

Hint: Find a constant $\displaystyle 0<q<1$ such that $\displaystyle |a_{n+2}-a_{n+1}|<q|a_{n+1}-a_{n}|$.