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equal products of real numbers

Source: Nordic Mathematical Contest 2002 #3

September 24, 2017

Productcombinatorics

Problem Statement

Let

{a_1, a_2, . . . , a_n,}

and

{b_1, b_2, . . . , b_n}

be real numbers with

{a_1, a_2, . . . , a_n}

distinct. Show that if the product

{(a_i + b_1)(a_i + b_2) \cdot \cdot \cdot (a_i + b_n)}

takes the same value for every

{ i = 1, 2, . . . , n, }

, then the product

{(a_1 + b_j)(a_2 + b_j) \cdot \cdot \cdot (a_n + b_j)}

also takes the same value for every

{j = 1, 2, . . . , n, }

.

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