Problem of the year 2016

Algebra Level 5

\[\begin{align} ax + by & = 3 \\ a{ x }^{ 2 } + b{ y }^{ 2 } & = 7 \\ a{ x }^{ 3 } + b{ y }^{ 3 } & = 16 \\ a{ x }^{ 4 } + b{ y }^{ 4 } & = 42 \end{align} \].

If \(a\), \(b\), \(x\), and \(y\) are real numbers satisfying the system of equations above, find \(a{ x }^{ 5 } + b{ y }^{ 5 }\).

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