Pre-RMO 2014/9

Natural numbers $k, l, p$ and $q$ are such that if $a$ and $b$ are the roots of $x^2 -kx + l = 0$ then $a + \frac{1}{b}$ and $b + \frac{1}{a}$ are the roots of $x^2 -px +q = 0$ . What is the sum of all possible values of $q$ ?

This note is part of the set Pre-RMO 2014

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Pre-RMO 2014/9

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