HMMT February 2022 #0koj

HMMT February 2022Hard

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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.

VectorsCauchy-SchwarzInvariants / monovariants

Problem

Problem: On a board the following six vectors are written:

(1,0,0), \quad(-1,0,0), \quad(0,1,0), \quad(0,-1,0), \quad(0,0,1), \quad(0,0,-1)

Given two vectors $v$ and $w$ on the board, a move consists of erasing $v$ and $w$ and replacing them with $\frac{1}{\sqrt{2}}(v+w)$ and $\frac{1}{\sqrt{2}}(v-w)$ . After some number of moves, the sum of the six vectors on the board is $u$ . Find, with proof, the maximum possible length of $u$ .

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