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There exists an irrational number r (IMO SL 1987-P23)

Source:

August 19, 2010
modular arithmeticalgebrapolynomialirrational numbernumber theoryIMO Shortlist

Problem Statement

Prove that for every natural number kk (k≥2k \geq 2) there exists an irrational number rr such that for every natural number mm, [rm]≡−1(modk).[r^m] \equiv -1 \pmod k .
Remark. An easier variant: Find rr as a root of a polynomial of second degree with integer coefficients.
Proposed by Yugoslavia.
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