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jmc

algebra intermediate

Problem

Let

m \circ n = \frac{m + n}{mn + 4}

. Compute

((\dots ((2005 \circ 2004) \circ 2003) \circ \dots \circ 1) \circ 0)

.

Solution

Note that

m \circ 2 = (m + 2) / (2 m + 4) = \frac{1}{2}

, so the quantity we wish to find is just

(\frac{1}{2} \circ 1) \circ 0 = \frac{1}{3} \circ 0 = \frac{1}{12}

.

Final answer

\frac{1}{12}