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Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be real numbers with $a_1, \ldots, a_n$ distinct. Show that if the product $\left(a_i+b_1\right)\left(a_i+b_2\right) \cdots\left(a_i+b_n\right)$ takes the same value for every $i=1,2, \ldots, n$, then the product $\left(a_1+b_j\right)\left(a_2+b_j\right) \cdots\left(a_n+b_j\right)$ also takes the same value for every $j=1,2, \ldots, n$.
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