<s>[INST] A set of N distinct points in a plane have the property that no three lie on the same line. Prove that one can assign a number, either +1 or -1, to each point in such a way that the sum of their products taken three at a time is 0.

For example, given 4 points, one way to assign the numbers would be +1 to the first 3 points and -1 to the last point. Then, the sum of their products taken three at a time will be (+1)(+1)(+1) + (+1)(+1)(-1) + (+1)(-1)(-1) + (-1)(-1)(+1) + (-1)(+1)(+1) + (+1)(-1)(+1) = 1 + (-1) + 1 + 1 + (-1) + 1 + 1 = 3(-1) + 4(1) = 1, but this is not 0. 

However, if we assign the numbers +1, +1, -1, and -1 to the 4 points in some order, then the sum of their products taken three at a time will be (+1)(+1)(-1) + (+1)(-1)(-1) + (-1)(-1)(+1) + (-1)(+1)(+1) + (+1)(-1)(+1) + (-1)(+1)(-1) = (-1) + 1 + 1 + (-1) + 1 + (-1) = 0.
 
Here we see that we can assign the numbers +1 and -1 to the 4 points in such a way that the sum of their products taken three at a time is 0.

But what about for any N points?[/INST]