ILL 1987 Problem 78

Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$, $$[r^m] \equiv -1 \pmod k .$$

Remark. An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients.

Proposed by Yugoslavia.