Question 27929: Suppose 5 points with integer coordinates are chosen at random from the xy-coordinate system. Show that the midpoint of a line segment that connects at least one pair of these points has integer coordinates.
Answer by bmauger(101)   (Show Source):
You can put this solution on YOUR website! Interesting problem. Think about the midpoint equation:
 
Note that you'll get integers when the top of each equation is an even number. So how do we get an even number from adding two different integers? When we add two even numbers, or when we add two odd numbers.
Odd+Odd=Even
Even+Even=Even
Odd+Even=Odd
So each ordered pair must be one of these four configurations:
(odd, even)
(even, odd)
(odd, odd)
(even, even)
If each of these were used, then each of the two midpoints would have at least one non-integer. However, when you add a fifth ordered pair, it must be the same configuration as one of those, and the midpoint must be an integer.
Example:
A(1, 2)
B(0, 3)
C(-1, -1)
D(4, -4)
Midpoints:
AB=(0.5, 2.5)
AC=(0, 0.5)
AD=(2.5, -1)
BC=(-0.5, 1)
BD=(2, -0.5)
CD=(1.5, -2.5)
So far, no midpoints that are both integers, but any ordered pair you pick now must have the same even/odd configuration as one of the prior, and the midpoint for the two will have to be integers.
Eg. E=(10, 5) (Even, Odd) same as B.
Midpoint BE=(5, 4) both integers.
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