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(a_2+a_3+...a_n)/a_1 x a_1+a_3+a_4+...a_n)/a_2...

Source: Austrian - Polish 1992 APMC

May 7, 2020
ProductSumalgebra

Problem Statement

Let n≥3n\ge 3 be a given integer. Nonzero real numbers a1,...,ana_1,..., a_n satisfy: −a1−a2+a3+...ana1=a1−a2−a3+a4+...ana2=...=a1+...+an−2−an−1−anan−1=−a1+a2+...+an−1−anan\frac{-a_1-a_2+a_3+...a_n}{a_1}=\frac{a_1-a_2-a_3+a_4+...a_n}{a_2}=...=\frac{a_1+...+a_{n-2}-a_{n-1}-a_n}{a_{n-1}}=\frac{-a_1+a_2+...+a_{n-1}-a_n}{a_{n}} What values can be taken by the product a2+a3+...ana1⋅a1+a3+a4+...ana2⋅...⋅+a1+a2+...+an−1an\frac{a_2+a_3+...a_n}{a_1}\cdot \frac{a_1+a_3+a_4+...a_n}{a_2}\cdot ...\cdot \frac{+a_1+a_2+...+a_{n-1}}{a_{n}} ?
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