If \frac{\log_{10}a}{b-c}=\frac{\log_{10}b}{c-a}=\frac{\log_{10}c}{a-b} (a \neq b \neq c), then what is the value of abc?

Explanation

Let each fraction be k. Then \log_{10}a = k(b-c), \log_{10}b = k(c-a), and \log_{10}c = k(a-b). Summing these gives \log_{10}a + \log_{10}b + \log_{10}c = k(b-c+c-a+a-b) = 0. Thus, \log_{10}(abc) = 0 \implies abc = 10^0 = 1.

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