HMIC #0jgc — HMIC · USAMO Guide - Olympiad Mathematics Learning Platform

HMIC #0jgc

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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

Problem:

A subset $U \subset \mathbb{R}$ is open if for any $x \in U$ , there exist real numbers $a, b$ such that $x \in (a, b) \subset U$ . Suppose $S \subset \mathbb{R}$ has the property that any open set intersecting $(0,1)$ also intersects $S$ . Let $T$ be a countable collection of open sets containing $S$ . Prove that the intersection of all of the sets of $T$ is not a countable subset of $\mathbb{R}$ .

(A set $\Gamma$ is countable if there exists a bijective function $f: \Gamma \rightarrow \mathbb{Z}$ .)