Problem 5

Let

P_1,P_2,\ldots,P_{1993}=P_0

be distinct points in the

xy

-plane with integer coordinates. Assume that no point other than

P_i

and

P_{i+1}

on the segment

P_iP_{i+1}

has both coordinates integers, for

i=0,1,\ldots,1992

. Prove that for some

i

0\le i\le1992

, there is a point

Q=(q_x,q_y)

P_iP_{i+1}

such that both

2q_x

and

2q_y

are odd integers.

Step 1 of 3: Classify integer points by parity

P=(a,b)\text{ is even or odd according as }a+b\text{ is even or odd}

Detailed analysis

Call an integer point $P=(a,b)$ even or odd according as $a+b$ is even or odd. There are only two classes.