IMO Team Selection Test #0jlv

IMO Team Selection TestHard

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Problem

For a prime $p$ , a subset $S$ of residues modulo $p$ is called a sum-free multiplicative subgroup of $\mathbb{F}_p$ if:

there is a nonzero residue $\alpha$ modulo $p$ such that $S = \{1, \alpha^1, \alpha^2, \dots\}$ (all considered mod $p$ ), and
there are no $a, b, c \in S$ (not necessarily distinct) such that $a + b \equiv c \pmod p$ .

Prove that for every integer $N$ , there is a prime $p$ and a sum-free multiplicative subgroup $S$ of $\mathbb{F}_p$ such that $|S| \ge N$ .