HMMT February #0k1a — HMMT February · USAMO Guide - Olympiad Mathematics Learning Platform

HMMT February #0k1a

HMMT FebruaryHard

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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:

Let $n$ and $m$ be positive integers which are at most $10^{10}$ . Let $R$ be the rectangle with corners at $(0,0)$ , $(n, 0)$ , $(n, m)$ , $(0, m)$ in the coordinate plane. A simple non-self-intersecting quadrilateral with vertices at integer coordinates is called far-reaching if each of its vertices lie on or inside $R$ , but each side of $R$ contains at least one vertex of the quadrilateral. Show that there is a far-reaching quadrilateral with area at most $10^{6}$ .

(A side of a rectangle includes the two endpoints.)