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Omni-MATH / On a board the following six vectors are written: (1,0,0), (−1,0,0), (0,1,0), (0,−1,0), (0,0,1), (0,0,−1). Given two vectors v and w on the board, a move consists of erasing v and …

Problem

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problem

On a board the following six vectors are written: (1,0,0),(−1,0,0),(0,1,0),(0,−1,0),(0,0,1),(0,0,−1)(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 vv and ww on the board, a move consists of erasing vv and ww and replacing them with 12(v+w)\frac{1}{\sqrt{2}}(v+w) and 12(v−w)\frac{1}{\sqrt{2}}(v-w). After some number of moves, the sum of the six vectors on the board is uu. Find, with proof, the maximum possible length of uu.
Plain-text mathematical notation (without MathML)
On a board the following six vectors are written: (1,0,0), (−1,0,0), (0,1,0), (0,−1,0), (0,0,1), (0,0,−1). Given two vectors v and w on the board, a move consists of erasing v and w and replacing them with (1)/(√(2))(v+w) and (1)/(√(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.
Original LaTeX notation
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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Source and history

task_id
47b116ca-62f0-57ea-8b8b-e2562344fc7b
task_key
test--47b116ca-62f0-57ea-8b8b-e2562344fc7b
task_revision_id
1
config / split
/ test

Official source

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