Let (xi; yi), i = 1; 2; 3; 4; 5 be a set of ve distinct points with integer coordinates in the xy plane. Show that the midpoint of the line joining at least one pair of these points has integer coordinates.

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- Mar 28th 2008, 06:06 PM #1

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- Mar 28th 2008, 07:13 PM #2
## The pigeonhole

Hi peiyilee,

The midpoint of the line connecting $\displaystyle (x_i,y_i) \text{ and } (x_j, y_j)$ has integer coordinates if and only if (1) $\displaystyle x_i \text{ and } x_j$ have the same parity (i.e., both are even or both odd), and (2) $\displaystyle y_i \text{ and } y_j$ have the same parity.

There are only 4 possibilities for the parity pair of $\displaystyle (x_i,y_i)$: (even,even), (even,odd), (odd, even), or (odd,odd). Since there are only 4 possibilities and you have 5 points, by the pigeonhole principle at least two points must have the same parity pair The line connecting these two points has a midpoint with integer coordinates.

jw

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