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sum \sqrt{a+\sqrt{b + \sqrt{c}}} <= 3\sqrt{m+\sqrt{m + \sqrt{m}}}

Source: Switzerland - 2007 Swiss MO Final Round p7

December 26, 2022
inequalitiesalgebraradical

Problem Statement

Let a,b,ca, b, c be nonnegative real numbers with arithmetic mean m=a+b+c3m =\frac{a+b+c}{3} . Provethat a+b+c+b+c+a+c+a+b≤3m+m+m.\sqrt{a+\sqrt{b + \sqrt{c}}} +\sqrt{b+\sqrt{c + \sqrt{a}}} +\sqrt{c +\sqrt{a + \sqrt{b}}}\le 3\sqrt{m+\sqrt{m + \sqrt{m}}}.
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