Mock AIME 6 2006-2007 Problems/Problem 8

Problem

A sequence of positive reals defined by $a_0=x$ , $a_1=y$ , and $a_n\cdot a_{n+2}=a_{n+1}$ for all integers $n\ge 0$ . Given that $a_{2007}+a_{2008}=3$ and $a_{2007}\cdot a_{2008}=\frac 13$ , find $x^3+y^3$ .

Solution

$a_0=x$

$a_1=y$

$a_2=\frac{a_1}{a_0}=\frac{y}{x}$

$a_3=\frac{a_2}{a_1}=\frac{\frac{y}{x}}{y}=\frac{y}{x}$

~Tomas Diaz. orders@tomasdiaz.com Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

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Mock AIME 6 2006-2007 Problems/Problem 8

Problem

Solution