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find irrational numbers such that these two things are rational

Source: Indonesia Regional 2023 Essay Number 4

July 16, 2023
irrational numberalgebraIndonesiaRMO

Problem Statement

Find all irrational real numbers α\alpha such that α3−15α and α4−56α \alpha^3 - 15 \alpha \text{ and } \alpha^4 - 56 \alpha are both rational numbers.
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