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Omni-MATH / In some squares of a 2012×2012 grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the g…

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problem

In some squares of a 2012×20122012\times 2012 grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle BB flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle BB. For all possible starting and ending configurations, find the maximum length of the sum of the [i]translation vectors[/i] of all beetles.
Plain-text mathematical notation (without MathML)
In some squares of a 2012×2012 grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle B flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle B. 
For all possible starting and ending configurations, find the maximum length of the sum of the  [i]translation vectors[/i] of all beetles.
Original LaTeX notation
In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle $B$. 
For all possible starting and ending configurations, find the maximum length of the sum of the  [i]translation vectors[/i] of all beetles.

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Source and history

task_id
1206e361-b869-5f0b-9431-1ec6c78e017a
task_key
test--1206e361-b869-5f0b-9431-1ec6c78e017a
task_revision_id
1
config / split
/ test

Official source

initial import