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sum 1/(a_i +2019)=1/2019, min of product a_1a_2... a_{2019}, a_i>0

Source: The 1st SAFEST Olympiad 2019 p4

September 9, 2019
minimumSumalgebrainequalities

Problem Statement

Let a1,a2,...,a2019a_1, a_2, . . . , a_{2019} be any positive real numbers such that 1a1+2019+1a2+2019+...+1a2019+2019=12019\frac{1}{a_1 + 2019}+\frac{1}{a_2 + 2019}+ ... +\frac{1}{a_{2019} + 2019}=\frac{1}{2019}.
Find the minimum value of a1a2...a2019a_1a_2... a_{2019} and determine for which values of a1,a2,...,a2019a_1, a_2, . . . , a_{2019} this minimum occurs
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