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Find the value of 10n where n is the minimum of an expression

Source: Bangladesh Mathematical Olympiad 2021 Problem 2

February 20, 2022
algebra

Problem Statement

Let u,vu, v be real numbers. The minimum value of u2+v2+(u−1)2+v2+u2+(v−1)2+(u−1)2+(v−1)2\sqrt{u^2+v^2} +\sqrt{(u-1)^2+v^2}+\sqrt {u^2+ (v-1)^2}+ \sqrt{(u-1)^2+(v-1)^2} can be written as n\sqrt{n}. Find the value of 10n10n.
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