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There exists at least one finite set T of positive integers

Source: Romanian Master in Mathematics 2009, Problem 4

March 7, 2009

trigonometryinvariantalgebra unsolvedalgebra

Problem Statement

For a finite set

X

of positive integers, let \Sigma(X) \equal{} \sum_{x \in X} \arctan \frac{1}{x}. Given a finite set

S

of positive integers for which

\Sigma(S) < \frac{\pi}{2},

show that there exists at least one finite set

T

of positive integers for which

S \subset T

and \Sigma(S) \equal{} \frac{\pi}{2}.
Kevin Buzzard, United Kingdom