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weights of all expansions

Source: Japanese MO 2003, Problem 4

April 20, 2007
number theory unsolvednumber theory

Problem Statement

Let p,q≥2p,q\geq 2 be coprime integers. A list of integers (r,a1,a2,...,an)(r,a_{1},a_{2},...,a_{n}) with ∣ai∣≥2|a_{i}|\geq 2 for all ii is said to be an expansion of p/qp/q if pq=r+1a1+1a2+1...+1an\frac{p}{q}=r+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{...+\frac{1}{a_{n}}}}}. Now define the weight of an expansion (r,a1,a2,...,an)(r,a_{1},a_{2},...,a_{n}) to be the product (∣a1∣−1)(∣a2∣−1)...(∣an∣−1)(|a_{1}|-1)(|a_{2}|-1)...(|a_{n}|-1). Show that the sum of the weights of all expansions of p/qp/q is qq.
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