HMIC #0jy8 — HMIC · USAMO Guide - Olympiad Mathematics Learning Platform

HMIC #0jy8

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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 $v_{1}, v_{2}, \ldots, v_{m}$ be vectors in $\mathbb{R}^{n}$ , such that each has a strictly positive first coordinate. Consider the following process. Start with the zero vector $w=(0,0, \ldots, 0) \in \mathbb{R}^{n}$ . Every round, choose an $i$ such that $1 \leq i \leq m$ and $w \cdot v_{i} \leq 0$ , and then replace $w$ with $w+v_{i}$ . Show that there exists a constant $C$ such that regardless of your choice of $i$ at each step, the process is guaranteed to terminate in $C$ rounds. The constant $C$ may depend on the vectors $v_{1}, \ldots, v_{m}$ .