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Algebra Difficulty 4.6 AIME Find the answer

Let m∘n=(m+n)/(mn+4)m \circ n=(m+n) /(m n+4). Compute ((⋯((2005∘2004)∘2003)∘⋯∘1)∘0)((\cdots((2005 \circ 2004) \circ 2003) \circ \cdots \circ 1) \circ 0).

A number or a short expression. Spacing and $ signs are ignored.

Solution

Note that m∘2=(m+2)/(2m+4)=12m \circ 2=(m+2) /(2 m+4)=\frac{1}{2}, so the quantity we wish to find is just (12∘1)∘0=13∘0=1/12\left(\frac{1}{2} \circ 1\right) \circ 0=\frac{1}{3} \circ 0=1 / 12.

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