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Equivalence with arctan

Source: RMO District Round, Bucharest 2008, Grade 10, Problem 2

January 27, 2008

functionalgebradomainalgebra proposed

Problem Statement

Consider the positive reals

x

y

and

z

. Prove that: a) \arctan(x) \plus{} \arctan(y) < \frac {\pi}{2} iff

xy < 1

. b) \arctan(x) \plus{} \arctan(y) \plus{} \arctan(z) < \pi iff xyz < x \plus{} y \plus{} z.