Let $p >5$ be a prime number. Prove that every algebraic integer of the $p$th cyclotomic field can be represented as a sum of (finitely many) distinct units of the ring of algebraic integers of the field.
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Miklos Schweitzer is a very hard contest.
Anyway, solution for this (and other problems) can be found in the book:
Contests in Higher Mathematics, published by Springer.
Google books has it:
And this particular problem's solution appears here: