, 2009

If $a > 0$ and $b > 0$, a new operation $\nabla$ is defined as follows: $a\nabla b = \dfrac{a+b}{1+ab}$.
For example, $3\nabla 6 = \dfrac{3+6}{1+3\times 6} = \dfrac{9}{19}$.
(a) Calculate $2\nabla 5$.
(b) Calculate $(1\nabla 2)\nabla 3$.
(c) If $2\nabla x = \dfrac{5}{7}$, what is the value of $x$?
(d) For some values of $x$ and $y$, the value of $x\nabla y$ is equal to $\dfrac{x+y}{17}$. Determine all possible ordered pairs of positive integers $x$ and $y$ for which this is true.