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Found problems: 1

2021 Belarusian National Olympiad, 10.8
Tags: algebra , cubic root
Two numbers $1+\sqrt[3]{2}+\sqrt[3]{4}$ and $1+2\sqrt[3]{2}+3\sqrt[3]{4}$ are given. In one move you can do one of the following operations: 1. Replace one of the numbers $a$ with either $a-\sqrt[3]{2}$ or $-2a$ 2. Replace both numbers $a$ and $b$ with $a-b$ and $a+b$ (you can choose the order of $a$ and $b$ yourself) Prove that the obtained numbers are always non-zero