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Looks like number theory but actually an inequality problem

Source: German TST 2004, Exam V, Problem 4, MOCP July 2000, pr. 21

May 1, 2004
GermanyTSTTeam Selection Testequationinteger equation

Problem Statement

Let the positive integers x1x_1, x2x_2, ......, x100x_{100} satisfy the equation 1x1+1x2+...+1x100=20.\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20. Show that at least two of these integers are equal to each other.
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