HMMT February

Algebra Difficulty 4.6 AIME Find the answer

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

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

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

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